Numerical Methods Project

Numerical methods provide a way to compute (approximate) values of solutions to differential equations, even when we cannot solve the equations exactly.  The drawback is the large number of numerical calculations required to obtain a desired value and level of precision.  In this project, you will use technology to implement the various numerical methods and use your technological solution to solve differential equations problems.

MAT 2680 Project

(Parts I-IV due Tuesday, April 13th)  Use technology to compute approximate solutions to the initial value problem below using Euler’s Method, the Improved Euler’s Method, and the Runge-Kutta Method.

Initial Value Problem. Given the differential equation dy/dx=2x-y and initial condition y(0.2)=1.7, find the value of y(0.8).

Part I: Use technology (see “Technology Options” below) to approximate the solution to the Initial Value Problem using Euler’s Method, the Improved Euler’s Method, and the Runge-Kutta Method, each with a step size h=0.05. See “Project Requirements” below for more details.

Part II. Find the exact solution y(x) to the Initial Value Problem and use it to determine the exact value of y(0.8). Round your answer to 8 decimal places.
Type or write your answer to this part and submit it with your project.

Part III. Compare the exact value of y(0.8) obtained in Part II to the three numerical approximations obtained in Part I. How many correct decimal digits did each method produce? Which method was the most accurate?
Type or write your answer to this part and submit it with your project.

Part IV. Reflection.  Write one or two paragraphs (minimum 300 words) responding to the following.  Leave your response to this part as a comment on this post.

  1. Describe your project and how it works.
  2. Describe the process of building your numerical methods calculator.  What kind of technology did you decide to use, and why? Did you encounter any unexpected challenges in completing this project?
  3. Why do we need numerical methods in addition to the other methods studied in the class?
  4. Why is this assignment included in the class (instead of just computing the various methods using a calculator)?
  5. Please include a link to your project (if it is online), or clearly state that you will be sending me the files by email (and don’t forget to do it!).

Technology Options

You can choose your technology tool for this job – use any one of the following:

  1. a spreadsheet (Excel, Google Sheets, or other spreadsheet)
    • if you choose to create a spreadsheet, you should have columns forx, y, f(x,y) , and so on, and each stage should appear in its own row (See Project Requirements below for more details)
    • HOW TO SUBMIT: If your project is a spreadsheet, either share it with me (if it is in Google Sheets or a similar cloud-based platform), or email the file to me as an attachment.
  2. a programming language (Java, Perl, or other programming language)
    • if you choose to write code, your program should output the values of x, y, f(x,y) and so on at each stage (See Project Requirements below for more details)
    • HOW TO SUBMIT: If your project is code, please submit it using an online coding site like ideone.com – once your code is working on the site, you can simply submit a link.  If you are using a programming language not supported by ideone.com, you can email the source code to me.
  3. mathematical software (MatLab, Maple, Mathematica, or other mathematical software)
    • if you choose to use mathematical software, your program should output the values of x, y, f(x,y) and so on at each stage (See Project Requirements below for more details)
    • HOW TO SUBMIT: If your project uses mathematical software, either share it with me (if it is in MatLab Online or a similar cloud-based platform), or email the file to me as an attachment.

Project Requirements

  • Your solution must be able to carry out Euler’s Method, Improved Euler’s Method, and Runge-Kutta (you may implement these as three separate spreadsheets or programs if you wish).
  • Your solution should display all the points (x,y) found along the way, not just the final point.
  • Your solution should also display other values found while carrying out each method:
    1. Euler’s Method: display the slope f(x,y) at each stage
    2. Improved Euler’s: display the values of k1, k2 at each stage
    3. Runge-Kutta: display the values of k1, k2, k3, k4 at each stage
    4. You can display other values as well, if you wish (for example, the intermediate y-value in the Improved Euler method that we refer to as z).
  • Your solution may NOT use any built-in version of these methods (for example, most mathematical software contains a built-in command for Euler’s Method – you can use this to check your work, but you need to create your own solution).

Test Your Project

Test your project.  Data for the first few stages of calculation using each method appear below (this data was generated using a spreadsheet – Google Sheets)

Euler’s Method

ihx_iy_ik = f(x_i,y_i)y_(i+1)
00.050.21.7-1.31.635
10.050.251.635-1.1351.57825
20.050.31.57825-0.978251.5293375
30.050.351.5293375-0.82933751.487870625

Improved Euler’s Method

ihx_iy_ik1z_(i+1)k2y_(i+1)
00.050.21.7-1.31.635-1.1351.639125
10.050.251.639125-1.1391251.58216875-0.982168751.586092656
20.050.31.586092656-0.98609265631.536788023-0.83678802341.540520639
30.050.351.540520639-0.84052063931.498494607-0.69849460731.502045258

Runge-Kutta

ihx_iy_ik1 = f(x_i,y_i)k2 = f(x_i+.5h,y_i+.5hk1)k3 = f(x_i+.5h, y_i+.5hk2)k4 = f(x+h,y+hk3) y_(i+1) = y_i + h*(k1+2k2+2k3+k4)/6
00.050.21.7-1.3-1.2175-1.2195625-1.1390218751.639057109
10.050.251.639057109-1.139057109-1.060580682-1.062542592-0.98592997981.585963496
20.050.31.585963496-0.9859634957-0.9113144083-0.9131806355-0.8403044641.540336345
30.050.351.540336345-0.8403363453-0.7693279367-0.7711031469-0.7017811881.501811514

28 Comments

  1. Showmik K. Chowdhury

    Part IV:
    In order to project for part one, I get to calculate the given equation with three different numerical methods. So, for this part, I picked excel to do the calculation for me. I did three files because I get to use three different methods. The first method that I used it called Euler’s Method. This method was simpler and shorter than the two other methods. In this following method, I started with a given step size of h = 0.05 which was the same throw-out the project. And there was given (xi, yi). since, (xi, yi) was given so I build my own formula with my known information to complete my (x) & (y) column. Then I used the value to calculate the (k) column. Then finally I build my final formula which going to calculate Y_(i+1) column.
    Similarly, for Improved Euler’s method, I followed the same technic calculated till (k1). Then for this method, I add two more extra columns known as (z) & (k2) to increase the accuracy of my calculation. And then I calculated Y_(i+1) column using all the column values.
    Finally, for the Runge-Kutta method, I followed the same step as Euler’s method until my (k1) column. Then I reached to (k2, k3, k4) column I bring up some change in my excel formula to get an even more accurate value for my calculation. Therefore, K1 formulate is different than (k2, k3, k4) formula. Even though I used the same formulate (k2, k3, k4) but their value was different than one another. Because I used the previous (k) value to calculate the next(k) value. Is it what unique about this method and all of those k values work as a filter and help us to get more accurate value possible from our calculation.
    Since I am familiar with excel and learn the material I did not encounter any problem to do this project. I believe that numerical methods are very important for real-life experience. Because those methods help us to get as accurate possible answers which helps us to get the perfect results in our real-life experience. For example, biologists could use this method to get very specific data possible which will help them to come up with better results for their experiments.
    I believe this assignment was included in this class to make us realized that how these materials are so important for us and how they could be used in real-life applications.

  2. Faisal

    For the project, I used one excel file and in that file, I used three different methods. The first method I did is that I used is called Euler’s Method. This method is kind of easy and simple than the other two methods. I am going to talk about it later. So the Euler’s Method we have given the step size of 0.05 which was the same for the other two methods since we were given xi and Yi. I made the formula in excel where I had to calculate for the column k. Then my final formula was to calculate Yi+1 Column
    So Far the improved Euler’s method I pretty much did the same thing then I calculated this time for the z and k2 column and then I calculated for Yi+1
    Then for the third method which is called the Runge Kutta Method that I had to Calculate for the k2, k3, and k4 column and keep in mind for all of them, I used to calculate for the k1 column and the k1 column is different than the k2, k3, and k4 column.
    I did not encounter any unexpected challenges but I did had a tiny problem where my formula answer was coming as a date like actual MM/DD/YYYY in order to fix that I had to change the settings in excel but after that i did had no problems
    Well, Numerical methods are very important because they help us in real life, and what I mean by that is, Scientist, Mathematician, etc people could get accurate and precise results on what they are working on it could be an experiment, in class, or anywhere and I also believe that we were supposed to do this project to realize that how important this is if you don’t have a calculator in hand with you!

  3. Mohammad Masud

    Mohammad Ishaq Masud
    03/25/2021

    Part IV: (Reflection)
    Working:
    This project is about the numerical approximation of the problems related to differential equations using Euler’s Method, Improved Euler’s Method, and Runge-Katta Method.
    There is a Worksheet named as inputs. Here, the required inputs i.e., Initial Values and step size, are filled. All the methods will be taking input from this sheet. Then we have implemented the different methods by simply apply their formulas to the columns. Their formulas are given below:
    Euler’s Method:
    y_(i+1)= y_i+h×f(x_i,y_i)
    Instead of applying the formula altogether, we have got some intermediate values,
    dy/dx=f(x,y)= 2x-y
    Improved Euler’s Method:
    y_(i+1)= y_i+h/2 (k_1i+k_2i )
    The intermediate values for Improved Euler’s Method are,
    k_1i= 2x-y
    z_(i+1)=y_i+h×k_1i
    x_(i+1)= x_i+h
    k_2i= 2(x_(i+1) )-(z_(i+1))
    Runge-Kutta Method:
    y_(i+1)= y_i+h/6 (k_1+2k_2+2k_3+k_4 )
    The intermediate values for Improved Euler’s Method are,
    k_1=f(x_i,y_i )=〖2x〗_i-y_i
    k_2=f(x_i+h/2,y_i+h×k_1/2)
    k_3=f(x_i+h/2,y_i+k_2/2)
    k_4=f(x_i+h,y_i+k_3 )

    Build:
    We used Spreadsheet Technology (Excel) because it does not need any programming language IDE or Mathematical software to be installed. Every person has Microsoft Excel or Google Sheets in their Computers, so it becomes more feasible for everyone to use this Approximation Calculator easily.
    Secondly, it is very easy to implement as we have to apply the formula of the method to the columns and get our answer. We used some intermediate values (Columns) to make it more simplified.
    It was very easy and simple to use MS Excel for this project but there is a drawback because f(x,y) is fixed, we have to change each worksheet in order to implement any other problem, while in a programming language, we can approximate any problem by just replacing the function definition part. But for this project, we have only one problem, so it was better to use Excel.

    Need of Numerical Methods:
    Many differential equations cannot be solved using an analytical approach. For example, y^’=x-y^2 cannot be solved analytically. Here numerical method helps to approximate the solution. Although the solution will not be exact or very precise, it will be close to the actual solution. The error caused by numerical methods can be omitted.
    Besides this, there are many differential equations that can be solved analytically but are very complex and takes a lot of time. To overcome this problem, we can use numerical methods to approximate the solution.

    This project teaches us how we do big calculations quickly and efficiently instead of long-hand calculations. This project has taught us how we can input three different methods in computer programs (Excel in my case) and get the answers. This a digital application of what we learned in class.

    I will be sending the files( Part 2 and Part 3) by Email.

  4. Micheal Lewis

    For this project I decided to use google sheets for my numerical methods project showing three different methods. These three different methods are called the Euler’s Method, Improved Euler’s Method and the Runge Kutta method. My project demonstrates a faster and more precise way of getting to a certain X coordinate to obtain a Y-coordinate using any step size. I decided to use Google Sheets because in highschool I took a finance class where my teacher taught me various ways how to use excel/sheets. By using my knowledge there I created three different tables for each method. I wrote down the step size h which was 0.05 and formatted the rest of the cells below it to have the same step size. Then I formatted the cells below for the n column to go from 1 to whatever number I needed to get to 0.8. For the X- coordinates I inputted a mathematical equation where the next cell below each other would add 0.05 (the step size) to the x- coordinate before. I manually inputted the initial values for X_0 and Y_0. To get k I put the equation given to us and found the number. I did this for all Z, k, and Y_(n+1) values. The way google sheets works is that once you do the same pattern once or twice you can drag the cursor down to get the rest of values. For this project I will send you the files for all parts but I will also share my google sheets with you so u can click on the cells to see the equations I inputted to get all my answers. I also checked some of my answers with my calculator to see if the answer google sheets gave me was in fact correct. During this project I encountered no issues with finding the values and maneuvering through excel. I believe that having this technique and knowing how to do it manually and technology is as important as solving differential equations. Maybe some differential equations cant be solved or is very hard to solve so by using the most accurate method the Runge-Kutta you can give an approximate answer. In class we used big step sizes but using smaller step sizes gives us more accurate answers. Using excel I am able to use the smallest step size I desire and let the computer compute to get a more precise answer instead of taking hours on my calculator to do 15+ rounds.

  5. Oscar Ramirez

    In this project I chose to use Microsoft Excel because the spreadsheet has a very interesting function to make this project do able and I’m more comfortable using Excel. I first open a file and open three tabs in the bottom and named them base on the method I would be using, those three being Euler’s Method, Improved Euler’s Method and the Runge-Kutta method. The good thing about Excel is that the spreadsheet uses boxes with columns and rows so already making the data look like it’s on a table so it’s easier to see what going on.

    The way this project works is similar to how we do it in class, using the table to keep track of our numbers, values, and using formulas to help fill in the table until we get our answer. Each column represents a variable like i, h, x_i, y_i, K, and y_(i+1). One column will be i and another column for h will be the step size of 0.05, and x_i will be our x-values starting with 0.2, which is given, until we reach 0.8 which is the x-value needed to find y at that point. To do this we can use the math function build into Excel to do the calculation for us, it won’t solve Euler Method for us but just do basic arithmetic operation. The equal sign is a command that will solve basic arithmetic equation you give it using the values from the spreadsheet.

    For instance, instead of adding 0.05 to each x-value manually on our calculator and then inputting the values into the Excel until we reach 0.8, we use the equal sign command (=). We click the box where the next x-value will go, and we write: =C4+0.05. This is an equation saying that we will use box C4 which has the value 0.2 for me on the spreadsheet and add 0.05 to it, once we hit enter that box will show 0.25 as the answer and if you click that box it will highlight the box and in the bottom right-corner of that highlighted box will be a dot where you can click and drag down the column and it will repeat the equation using the last x-value until you reach 0.8.

    There was no major problem only when I got to the Runge-Kutta method I needed more columns to represent different variables/values for K’s because each K use different formulas and x-values, so I use more columns to find those values and then implement them into the K equation to make Excel solve the equations easier instead of making a longer equation.

    The reason we need numerical methods in our studies is because while we can solve a differential equation normally but for some occasion sometimes its not possible to solve the differential equation so we apply the numerical method to solve the equation and give us an approximation of what the answer can be using step size, h.

    This assignment was included to show us that there is easier way to get the solution while we can use a calculator to solve this problem it would take a lot of calculation to achieve the solution. The use of computer software to do the work for us does help and even give a better approximation of the answer. Also, this gives us a view of real-world application some people in the work field won’t have enough time to calculate the answer with a calculator but if they have a program that they use in everyday work they can program the computer to solve it for them to get a better result.

    I will be sending the Excel file and the written response for PART II and III by email.

  6. Mahadi Sakib

    My project utilizes Excel software where it calculates each value right after inputting the correct equations. Excel is programmed where when given certain parameters it will continue to follow those parameters if it is given the correct values. For example, when working in excel I needed to make sure to get yi+1 for the initial condition then yi+1 for the next one. After Getting the next y values for these two points, I could use excel to automatically calculate the rest of the values since it understands that I want it to follow the same parameters I used to calculate the values for the first two points. I did not have much difficulty because I am familiar with excel, especially having to work with excel from previous classes that required something like this project (i.e., labs). Numerical methods help where when one method fails, we can refer to another, it helps to have these methods to provide at least an answer. This assignment was included in this class for students to experience and familiarize themselves with the similarities that come from these methods and how close they come to the exact value. It teaches us, ‘here is this other method you can use that comes close to the exact value which you could not get from the other methods.’
    I will be sending my project through email, one is an excel spreadsheet while the other is a text document.

  7. Jeremy Babb-Edwards

    For my project on numerical methods, I used google sheets to solve the given differential equation. I created three different charts. The first chart implemented the Euler’s Method. The second chart was completed by the Improved Euler’s Method. The third and final chart was the Runge-Kutta Method. For the Euler’s Method I displayed the slope f(x, y) at each stage. The Improved Euler’s Method showed the values of K1 and K2 at each stage. The Runge-Kutta Method gave the solutions of K1, K2, K3, and K4 at each stage. The way this works is by adding the formulas into each column. For example, in the Euler’s method, to get y(i + 1) = yi + k * h, I would have to implement the formula as y(i + 1) = D2 + E2 * B2. Building this numerical methods calculator was simple using google sheets. There were no major problems, except for a calculation error when finding the exact solution. We need numerical methods in addition to other methods because in certain cases, we cannot get the exact solutions to differential equations. Numerical methods allow us to get close approximations of solutions. I believe this assignment was included in the class, to show us that there is more than one way to compute our solutions, instead of just using a calculator. Creating a spreadsheet or program allows us to make more numerical calculations in a shorter time.

    Exact Value: y(0.8)= 1.41107839

    The Runge-Kutta method was the most accurate.
    The comparisons are under the Runge-Kutta spreadsheet.
    I will email you the other parts.

    Euler’s Method: https://docs.google.com/spreadsheets/d/1ZM6qDNlaZ70CXh073Yl3YoiZcoX8nusir95zDfCVI8g/edit?usp=sharing

    Improved Euler’s Method: https://docs.google.com/spreadsheets/d/193Ar52-mCj5dk8-_2ar4PY5MEMPlBsLQgVJRcl8arSo/edit?usp=sharing

    Runge-Kutta Method: https://docs.google.com/spreadsheets/d/1CEAXX-EBVHXuGj-avMAjzHucH2GSgM1m0KGFiYSBukU/edit?usp=sharing

  8. Jiefeng Huang

    In this project, I chose to use Microsoft Excel. Because I am not familiar with MATLAB or use a programming language to create calculators. I made three different excel spreadsheets, one for each method. The setting method of each spreadsheet starts with the initial x_i and y_i values, but the x_i starts from 0.2 and adds 0.05 every time. For these three methods, I use a step size of 0.05. The first table is Euler’s method is the easiest method because it only requires two formulas. One formula calculates k=f(x_i,y_i), and the other formula calculates y_(i + 1)=y_i+k*h.
    The second spreadsheet is Improved Euler’s Method is slightly more complicated, because it involves more formulas, and then the x_i and y_i of this table are the same numbers as the first spreadsheet. But x_i starts from 0.2 and adds 0.05 every time. In addition, Improved Euler’s Method needs to calculate K1, z_(i+1), K2, and y_(i+1). The formula is k1 = 2x-y, z_(i+1) )=y_i+h*K1,k_2i= 2*x_i-z_(i+1), and y_(i+1)=y_i+(K1-K2)/2*h.
    The third spreadsheet is Runge-Kutta which is the most complicated. Compared with the other two methods, this method incorporates more steps. The x_i and y_i are the same numbers as in the previous table. In this table, I need to calculate K1=f(x_i,y_i), k2 = f(x_i+.5h,y_i+.5hk1), k3 = f(x_i+.5h, y_i+.5hk2 ),k4 = f(x+h,y+hk3) and y_(i+1) = y_i + h*(k1+2k2+2k3+k4)/6. In short, these three spreadsheets use the initial x value and 0.2 I need to find all x values until it reaches 0.8. But, when I calculate Runge Kutta, I was paying attention to whether these numbers are mistyped, because it has a lot of decimal points, and if I make a mistake, the following numbers will be wrong. But the tragedy still happened, there were a few times when I type wrong numbers and when I finished comparing with the number I got from y (0.8) is not accurate. Then, I have to find out the wrong number and recalculate.
    The numerical method is very important because it can help us in real life. For example, it can use exact methods to cause errors on actual weather forecast problems. I believe this assignment has been included in our curriculum to help us better understand each numerical method.
    I will be sending the Excel file and PART II and III by email.

  9. Mahedi Bhuiyan

    For my project, I picked the excel version since it seems like the easiest one to me out of all of them. To find f(0.8) in excel, you can manually apply the exact equation to find whatever you are looking for. For example, to find Y1 in Improved Euler Method I had to apply this following code =D3+(E3+F3)/2*$B$3. This is the same equation as y_1=y_0+h/2(k10+k20). To find K1 I used =2*C3-D3 which is same as F(x0,y0). Since you start from 0.2 and you must find 0.8 with a step size of 0.05, that doesn’t mean you have to manually solve equation each time in excel. You can just solve one and for the rest just drag down the cursor until 0.8 and everything will automatically be calculated. I didn’t really encounter any problem other then just trying to figure out how to implement the actual math equation in excel. I think, Numerical Method is important because it gives us a close or precise answer to the actual exact answer. This project taught me the importance of different methods and how exact numbers are calculated.

  10. Kymora

    The introduction of the project provides our initial value problem as well as the given initial condition. The given initial condition is y’ = 2x – y and the initial condition is y(0.2) = 1.7. In order to complete the first part of the project, the three numerical methods would have to be used. In order to complete all numerical methods, I used the excel spreadsheet application. Each method has to use the size step of 0.05. The first method requested is the Euler’s method. The first column is i and it is followed by h. H is the step size which remains the same value. The third column is xi which can be determined by adding the step size starting from the value given, which is 0.2. Column yi follows the previous column xi, it can be determined by yi+1 starting with the value given 1.7. Column k is the slope, which can be found with the formula k = f( xi, yi). The last column is yi+1. The formula for finding its value is yi + h*k. For improved Euler’s formula, the steps are the same until after the slope , k, is calculated which is now k1. Column z(i+1) follows k1 and it is determined with formula yi+ h *k1. Column k2 is determined after, with the formula k2 = f(xi+1, zi). Then column yi+1 is found with formula yi + (k1+k2)/2*h. For the Runge-Kutta method, everything is the same until after k1. The next three columns are k2, k3, and k4. The formula for k2 is f(xi+(h/2), z2). The formula for k3 is f(xi+(h/2), z3) and k4 is f(xi+h, z4). yi+1 is followed and its formula is yi +(k1+3k2+2k3+k4)/6 *h.
    The exact solution to the initial value problem is required to be identified in part two. Once the exact solution is found, it is used to find the exact value of y(0.8). Part three is for comparing the numerical method results with the result found in part two of the project.
    Numerical methods are important because many differential equations cannot be solved exactly. By using numerical methods, we can obtain a value that is as close to the actual exact solution as possible. I will be sending you the finished numerical method calculations done on excel through email as well as part 2 and 3 of the project.

  11. Gianfranco Espinoza

    This MAT 2680 project consisted of creating a calculator of some sort that would solve the differential equation y’ = 2x – y. To do this I created three different programs using Visual Studios, each one using a different Numerical method.

    Each program was complete some what similarly, using for loops and variables that increment as the loop continued. There were no real issues other than the occasional error due to missing a semicolon every now in then. The process, though simple, took a consistent amount of focus in order to be able to identify the right values and to fix any errors that any arise when run the code. For instance, during the completion of the third program the first three values did not match the values that were given due to the equation being entered incorrectly.

    Most numerical methods are very simple to incorporate into differential equation however in return the numerical methods will retrieve a value that is not exact to the answer you seek. This is one of the down sides to using a numerical method instead of an exact solution however it can be noted that it requires less time and effort to complete and can allow the user to have a physical representation of the problem using plot graphs with the solution found throughout the numerical methods.

    As for why this topic is included in this course, I believe this is due to the idea that expanding the knowledge of efficiency in any form for any topic is important for the real world. This is similar to functions within C++, though it is not needed and it would probably be more effective to write out the code a function allows future projects to be more efficient.

    I will be sending the code through email

  12. julio dominguez

    For our project we where assigned to create a calculator so solve the differential equation y’=2x-y at the given initial condition of y(0.2) = 1.7 with a step size of 0.05 and goal of finding the value of y(0.8) = ?. The task is to create 3 different calculator to solve the initial value problem with the Euler’s Method, Improved Euler’s Method and Runge-Kutta Method.

    I chose to do this project in excel because I’m most familiar with this software and I have used excel extensively at my job. I first begun by creating 3 separate sheets and labeling them Euler’s, Improved Euler and Runge-Kutta Method respectively. I knew before starting the project that the three methods will be somewhat similar and the completion of the first sheet will help tremendously in completing the next. I then made a table with 8 columns titles i, h, xi, yi, k=f(xi, yi), y(i+1), k round & y(i+1) round. h is the step size which is given as 0.05 so I just wrote that in, xi is calculated by adding 0.05 to the initial condition of 0.2. Yi is calculating by starting with initial value of 1.7, the rest of the column can be calculated by adding the step size and multiplying it by the slope (k=f(xi, yi)). Next is the slope as mentioned before followed by y(i+1) which is calculated by adding yi with the slope and multiplying it by the step size of 0.05. I then used the round command to round all important columns to the 8th decimal place as stated in the project requirements. The hard part now is done, using excel you can automate the calculations by calling for specific cells, for example for my slope I calculated it by writing 2*C7-D7, C7 and D7 are both separate cells that I’m calling upon to complete the equation. Writing it out in this format makes it extremely easy to duplicate because to solve the next next row I can use Excels auto complete by highlighting a row and dragging it down, this replaces C7-D7 with the next cell of C8-D8 and so on solving every row for me. Using these strategies I was able to solve the Euler’s Method sheet. The main issue that ran into was when using excel to automatically complete the following rows. At first I highlighted the first two rows and dragged it down but excel seemed to get confused with the first row since I typed the given information in instead of using a formulas to solve them. After realizing my mistake I highlighted only the second row and dragged it down, this seemed to solve the issue until I saw that Xi stayed at 0.25 instead of increasing by the step size of 0.05 so I personally filled in the Xi column to get around this issue and saw that this solved all remaining problems with my calculator. Using the above method I solved the improved Euler’s method in a very similar fashion, I just needed to add a couple more columns to account for k1 and k2. In the Runge-Kutta method I also needed to add two addition columns for k3 and k4 but solved it using a similar method as the previous two sheets.

    The reason that we need numerical methods is to make our lives easier. I say this because they are many differential equations that are complicated if not nearly impossible to get the exact answer. Using the numerical methods we can approximate the answer in a much more simpler manner without going into the extensive work that is required to get the exact solution. I believe this project was assignment to better understand how numerical methods work and how to solve them, it is better to understand something when we apply the steps in different formats, in this case I used excel to better understand numerical methods.

    I have Emailed you my Excel Document with my project.

  13. Risul Rashed (Raysul)

    When I first read the project, my initial thought was that it would be very hard to finish. There were many different calculations involved and slight mistake can cause very different results. As soon as I started the project, I realized that calculations were very easy to compute, since most of the work is done the technology, we are allowed to use. At first, I thought of using MATLAB for my calculations. When I tried finding solutions for Euler’s Method with little code on MATLAB, I got some error which I couldn’t understand. Moreover, I couldn’t get it to show all the solutions in organized way. So, I dropped the idea of using MATLAB and gave Microsoft Excel a try. First, I solved Euler’s Method on excel, with given equation y^’=2x-y with initial x value of 0.2 and initial y value 1.7. I also know that I have to find solutions up to when x is 0.8 and step size is h=0.05. First, I find the k which is plug in the previous x and y value to equation (2x-y). After finding the k then solve for next y value by plugging in all the information to this equation y_(i+1)= y_1+h*k. Record all the values on a table and saved it. After that I created a new document to solve for improved Euler’s method. For this method I need to find k_1 which is plugging in x_i and y_i to the equation. Then find k_2 which is plugging in f(x,y) where x is x_i+h and y is y_(n-i)+h*k_1. After that I plugged in all the found values to the equation y_(n+1)= h/2*〖(k〗_1+k_2) which gives next y value and for next x value I just use x_n= x_(n-1)+h. Then solve it for all the y values until x is 0.8. After that I open another excel spread sheet and calculate using runge-kutta method. First find the next x_i using this x_i= x_(i-1)+h equation. Then find k_1, which is basically plug in x and y to the given equation. Then I find k_2 by plugging in f(x,y) to the equation, where x is x_i+h/2 and y is y_i+h/2*k_1. After that find k_3 which is also plugging f(x,y), but this time y is y_i+h/2*k_2. Then find k_4 by plugging in f(x,y) and y is y_i+h*k_3 this time. After finding all four k, I plugged in all the information to y_(i+1)=y_i+(k_1+2k_2+2k_3+k_4)/6 h. Which gives us next y. Using similar methods in excel I find all values.

    The reason we are using numerical method is from my understanding that we might not be able to find exact values for the original equation. In our case the equation was easy, and we were able to find the original equation by using nonhomogeneous procedures and get exact values. There will be a case where finding original equation is not possible, so that’s why we are using numerical methods.

    The reason we are solving numerical method using technology is because there are various different calculations with tiny details. If we missed a single calculation using a calculator and solving on paper it will take us tremendous time to find results. To save time and find results efficient way, we are assigned assignment this way.
    (I will email you all the works with subject: “Rashed’s Numerical Methods Project: OL78”)

  14. Saikat Babu

    At first, I chose to use MATLAB codes, but I got stuck in the middle of working. So, I had to change the Technology options (I will send you the code with the project). When I compared the value, I got in part II to Part I, it seems like Runge-Kutta Method in part I was the most accurate. Because runge-kutta methods we must do much more steps. Which give more accurate numbers. The first sheet of the excel chart carried out the Euler’s Method. The 2nd sheet is the Improved Euler’s Method, and the final sheet showing the Runge-Kutta Method. For the Euler’s Method I displayed the slope f (x, y) at every stage. The Improved Euler’s Method confirmed the values of K1 and K2 at every stage. The Runge-Kutta Method gave the options of K1, K2, K3, and K4 at every stage. The way this works is through including the formulation into every column.

  15. justin Bartholomew

    For this project we were told to create equation calculator with the methods that we learn in class. The Euler’s method, improved Euler’s method, and the Runge-Kutta Method were the methods assigned for this project and I decided to create my calculator in Microsoft excel. My main goal for this project was to take the initial differential equation of Y’=2x-y and solve them for each method.
    The reason why I chose to use excel was simply because even though I do know other coding programs such as matlab, java, and C++ I stuck to a program I felt more comfortable using. Plus creating my table for output was a lot easier to set up with the spread sheet that excel gives you. For me to complete this project I had to remember all the formula for each method used. I also decided to use the same table template that you gave to us. When creating this table I noticed that the differential equations problem gave us some hints when building our calculator we know that our step size (h) is .05 and our starting point on our table for Y(i) is .2 and ends at .8 so plus the step size tells us Y(i) is increasing by a increment of .05. Next thing I did was start work Euler’s Method I knew that the equations I would need is to start off by finding the formula for k=f(xi,yi)=2*xi-yi after that I then started the for the final column y(i+1)=yi+xi*h. For the improved Euler’s I know that there needs to be a K1 and K2 formula so I proceeded to plug these formulas into my cell on my spread sheet for improved Euler’s method K1=2*Xi*Yi then the next formula I need to solve Z(i+1)=Yi+K1*h afterwards I moved on to K2=2*Yi-Z(i+1) and finally I used the formula for Y(i+1)=yi+((K1+K2)/2)*h. Finally for the runge- kutta method I have to add K3 and K4 and use a completely different set of equations to figure out column K2 to column y(i+1) I start with K2 since the formula for K1 stayed the same as the previous improved Euler’s method. K2= 2*(xi+0.05*h)-(yi+0.05*h*k1) then K3=2*(xi+0.05*h)-(yi+0.05*h*k2) afterwards K4=2*(xi+0.05*h)-(yi+0.05*h*k3) and finally y(i+1)=yi+h*(K1+2*K2+2*K3+K4)/6 I used each one of these formulas in my top cells in order to fill each one of my columns for each project.
    The reason why the numerical method is important and should to be taught in other class is just for the main fact that solving differential equation is very long and tedious by using this method you can shorten you time for solving problems dramatically. The reason why I think this method was included in this class is because we need to understand how an equation like would come about. So as students we are finding out how the background behind differential equations works and not just take us straight to the answer.
    I have sent you my project on a excel file from my school email

  16. Erika

    I ended up using Excel since it seemed like the easiest route to take. But boy was I wrong! Since most of VIP information was given, I went on a YouTube binge to further my knowledge in Euler’s Method and in Excel. It took me a while but I believe I setup each workbook correctly. Each one is for each method taught in class. Why do we need numerical methods in addition to the other methods studied in the class? I ask myself this question everyday. (Just Kidding!) Some equations there is no exact solution to be found but an answer is still needed. These numerical methods help give an answer that is close is in approximation value that it is useable. This assignment included in the class to give students a reinforcement lesson on these methods since they are quite difficult to grasp. I have emailed a copy of the excel workbook to you.

  17. Chris Berry

    1. For this project, I decided to use excel, the project works inputting the initial conditions into the Excel Data table, doing this I was able to input the equation into the respective column to solve for the variable. Once once complete row was solved, I was able to Overwrite each cell with the equation in each respective column. This allowed me to simply only input the y-coordinate to its respective x-input, this would solve entire round and row.
    2.The process for solving for these Numerical methods, was to first solve for each Numerical Method on paper, this gave me a general consensus for the answer I would be looking for. The challenges I faced were mathematical, many of these Numerical Methods would take quite a while to solve, so any mishaps in the calculation would offset all of the further calculation and this was something that would happen to me often, it took me a while to get the hang of solving with Numerical Methods and Improved Numerical Methods, it almost feels like second nature, except for Runge Kutta which I don’t expect to be second nature for anyone.
    3.Numerical methods allow us to solve for accurate solution to difficult problem such as working with derivatives. Many mathematical problems are hard to solve directly, derivatives for example are hard to solve for at a specific point, but this allows to solve for these type of difficult problem..
    4.I feel like this assignment was included in this class because number 1, doing this with a calculator seems extremely difficult with so many numbers. Most importantly, however, I feel like this assignment was included because this type of course wouldn’t be taken in Majors other than engineering and science. This would be something, at least in my opinion that we should know for a future career.
    5. I will be sending my project via email

  18. Hameed Akinwunmi

    My project was made to be a calculator that can solve differential equations problems. The problem I had to solve “ y’=2x-y at the given initial condition of y(0.2) = 1.7 with a step size of 0.05 and goal of finding the value of y(0.8) = ?”. I Have to create 3 different calculators, one to solve Runge-Kutta Method, another is the initial value problem with the Euler’s Method, and lastly the Improved Euler’s Method.
    After looking at all the options I decided to use excel, because it’s much easier. I first had to make a table and fill in the variables after the variables have been put in you then have to put in the equations for example to find my second k i had to use this equation which integrates the numbers assigned to the positions.=2*K5-L5 so essentially the code/equation is telling excel that my second k is equal to 2 times the number located at K5 – L5. I did this process till all my numbers were filled in. I then repeated this process for the other two parts. I also double checked to make sure my first couple of numbers matched up to the ones you had already shared with the class.
    We need numerical methods and other things taught in this class because of the accuracy they can give us for example people who need to calculate things like how long a substance takes to get hot or cold. Scientists and mathematicians can use these methods to do long equations/large equations without errors. we are smart beings but like we say “everyone makes mistakes”, these help us limit the mistakes.
    I believe this assignment was included in this class to show us how to integrate skills that we already have to solve problems that we may have in the future. My project will be sent via email

  19. Edwin Urgilez

    For this project I used Microsoft Excel. I was and felt more comfortable using Excel than the other programs because I have little experience with the others. In this project I made three different spreadsheets, one for each method. All of these methods start of the same, such as the initial x_i is 0.2 and the initial y_i is 1.7. The step size is 0.05. The first spreadsheet was Euler’s method. This was the easiest one because it didn’t have a lot of formulas to use and therefore was the fastest to complete. There were only two formulas to calculate for, which were k and y_i+1. The formula for k would be f(x_i,y_i)= 2x-y. The formula for y_i+1= y_i+(h*k). The second spreadsheet was the second method which was Improved Euler’s Method. This method was a bit harder because it had more variables and formulas to use and input. So the extra variables would be z and there were two k’s instead of one. So it starts of with x_i 0.2 and each step size would be adding 0.05 to the previous step. It starts off the same in which you have to calculate for k1 first and the formula is f(x_i,y_i)=2x-y. Then calculate for z which is y_i+(k1*h). After that calculate for k2 which is f(x_i+1,z)=2x-y, x_i+1 would be the next step size x to use and the recent z calculation would be the form of y in this case . Finally calculate the last step for the next y interval, y_(i+1)= yi+((k1+k2)/2)*h. The third spreadsheet was the last and final method which was also the hardest due to its many formulas and extra variables to input. Extra as in k1, k2, k3, k4. But the way I did it was that i added extra variables such as ki(x_i,y_i). So for each k it consisted of different x and y variables every time and i thought it was easier to make a separate column’s for each k interval and that way just inputting the x and y’s into formulas. So to start off I first found k1 which was f(x_i,y_i)= 2x-y. Then I first found the x and y values for k2 to input the answer into the formula, k2(x_i)=x_i+(h/2) and k2(y_i)=y_i+(h/2)*k1 and got the reuslts and input to 2x-y. Next I did the same thing for k3. The k4 step consisted of a different formula which was k4(x_i)=x_i+h and k4(y_i)=y_i+(h*k3). The last step was to input all the k values and h to find the next y value, y_i+1= y_i+((h/6)*k1_i+(2*k2_i)+(2*k3_i)+(k4_i)). I believe we need numerical methods because while we can solve a differential equation normally but for some there are many occasion’s where it is not possible. We then apply the numerical method to solve the equation and it then give’s us an approximation of the answer. This assignment has been included in our class to help us better understand numerical methods. I will be sending the Excel file and Part II and III in PDF through email.

  20. Alberto Garib

    For this project I used Visual Studio IDE to write the program on C++ language. I decided to do C++ because it is the programming language that I feel the most comfortable to use from my programming background. Also, I feel that by taking this challenge I can help myself to get some practice along the way to build this program. The processes to build this project at first it became a somehow difficult at the beginning, but once the variables and the equations were set everything else started falling into place. For example, once the calculator for Euler’s Method was completed, the other methods just build into each other. One of the most difficult part for this program was to make sure that the console displayed the output into a table grid, this part took a lot of patient. The process to build this calculator is a little challenging because it requires a lot of critical thinking to set the program to focus on the function and keep the calculations at the require range. Moreover, I tried to keep the program as simple as possible and only use iteration to pass the value instead of increasing and move on to the following line until the loop reached end, this way the computer could automatically output each result on the console. The program was set so it could work with any number initial condition or steps and no just static solutions by setting the console to asks the user to enter the initial values of “y”, “x”, “the value of y(x) to find it, and at last “h”; after all of the numbers are entered, the program then add the values into the formulas already in the program and display the calculations on the console. We use numerical methods to study the approximation of errors of a numerical solution such as a differential equation even that we can see how some methods like the Euler’s method have a very big margin of errors compared to the Runge-Kutta method. By doing this project we can test the power of computer algorithms and see how it works.

    E-mailing the material.

  21. Harold

    My project is given the differential equation dy/dx=2x-y and having the initial condition of y(0.2)=1.7 I need to find the value when x=0.8 using Euler’s method, Euler’s improved method and the Runge-Kutta method. To do this I chose Excel since I’m more familiar with it than any other spreadsheet, programming language or mathematical software. First for the Euler’s method I created a column with the number of n values then another one for the step size h then another one for x then one for y, then one for the slope k and one for the new y value. For the Euler’s improved method I used the same thing except I used another k value and a z+1 and for the runge kutta I used 4 k values and 3 z -values z-2, z-3 and z-4. I then filled the 1st 2 or 3 rows using the formulas for each method and referring to each cell in Excel as an example C3 or B2 I entered the formulas like that and then I selected the first rows, highlighted and then clicked the plus button and dragged it towards the end and Excel automatically filled the other rows. I encountered 2 challenges doing this project because even though I’m mostly familiar with Excel I’m still not very familiarized. One challenge was that I had to look at you-tube videos to see how to put a formula on Excel, another one was that I didn’t know how to increase the number of decimals although it was pretty easy after I found out how to do it.
    We need numerical methods in addition to other methods studied in this class because we don’t always know the solution to a differential equation and there are many differential equations that we don’t know how to solve it and that’s where numerical methods come in handy so that we can approximate solutions that are very accurate like the Runge-Kutta method. This assignment is included in the class so that we get to really practice numerical methods so that we can understand it fully by doing a project about it because numerical methods are really important and we don’t know what situation will come later in our lives where we need to apply what we learned. I sent the project to your email.

  22. Michael Toussaint

    With the help of a classmate and class notes, I was able to use Excel and apply Euler’s Method, improved Eulers method, and the runge Kutta method to display the slope (x,y) at each stage display the values of k1,k2,k3,k4 at each stage and as well as the intermediate Y value in the improved Euler method. To do this I set values of X_0 and Y_0. To get k, I set the equation given to us and use excel to put the appropriate values. I did this for all Z, k, and Y_(n+1) values using the original formula. When reaching the (k2, k3, k4) column I had to change my excel formula to get a more accurate value for my calculation. Therefore, K1 formulate is different than (k2, k3, k4) formula. Even though I used the same formulate (k2, k3, k4) Their values apppered to be different this shows that the results may vary when conduction experiments and solving everyday problems, these methods are important because scientists, engineers and math maticians need this in order to get better and more accurate results, the intention is to improve data and technology when conducting experiments.

  23. Michael Toussaint

    With the help of a classmate and class notes, I was able to use Excel and apply Euler’s Method, improved Eulers method, and the runge Kutta method to display the slope (x,y) at each stage display the values of k1,k2,k3,k4 at each stage and as well as the intermediate Y value in the improved Euler method. To do this I set values of X_0 and Y_0. To get k, I set the equation given to us and use excel to put the appropriate values. I did this for all Z, k, and Y_(n+1) values using the original formula. When reaching the (k2, k3, k4) column I had to change my excel formula to get a more accurate value for my calculation. Therefore, K1 formulate is different than (k2, k3, k4) formula. Even though I used the same formulate (k2, k3, k4) Their values apppered to be different this shows that the results may vary when conduction experiments and solving everyday problems, these methods are important because scientists, engineers and math maticians need this in order to get better and more accurate results, the intention is to improve data and technology when conducting experiments.

  24. zcruz22

    For this project I decided to use Excel because I already use this program at work for many different calculating methods. I am quite comfortable with this program and find it very user friendly. Anyone can go to the spreadsheet and see what has been done. If you want to see how the result was obtained, all you have to do is double click the cell you wish to know more about.
    I created three separate sheets within my same file. Each one is labeled. The first one is Euler’s Method. In this spreadsheet I created a table similar to how we did in class. I started with an “i” column and went from zero to 12 down, since these are basically the amount of rounds we have to go in order to reach an x value of 0.8. The next column is called xi. This is where the x values go. We start off with an x value of 0.2. For the next x value going down I used the equal sign. We need to do this for everything we want Excel to compute. I did an equal and selected the cell with the 0.2 value + my h value of 0.05. I wrote my h value above separately from my table because this value will not change and I will be always using that same step size. In Excel we need to add a $ sign in between the cell name so that it locks the value and stays constant. If you don’t do this then the moment you drag down to fill the rest of the table, the h value will change and it will not work. So I was able to drag my x value down until it reached 0.8. Next was my y value column. We were given 1.7 value to start with. We had to find all the remaining y values, but those would have to wait until I finished setting up the first row completely. Next column is my k value. This is done by using the given equation. I put the equal sign and wrote 2*0.2(I clicked on the cell block this value is located) minus y(also by clicking on the cell box this is located) and enter. The value was given. Next column is what my next y value will be. yi+1. This is done by writing equal and clicking on my yi box, then + my k value (clicking on the k value box) * my h value located in another cell box and remembering to add the $ sign so it locks the step size in place.
    Once I have that value I go to the next y value and just write = and click the previous box where I got the new y value so it copies over to the next one. Then from here all I have to do is click on the lower right hand side of each cell box and drag it down up to the 0.8 row. This gives all the values and saves a lot of time. I also included a graph of the x and y to see how it would look.
    The next sheet is Improved Euler’s Method. This follows the same pattern as my previous sheet. For this method however I added a z column and a k2 column . The formula and strategy is different than Euler’s and I found it to be closer to the actual answer than Euler’s. To find the z, I used y+h*k1(my k1 is the same as with Euler’s) The next column was k2 and for this you use the given differential equation 2x-y. So I wrote = 2* the next x value of 0.25 (by clicking that cell) – y (for this I used my z value found previously). The next column is what the new y value will be. I used yi+k1+k2/2*h all by writing = and clicking the respective cell and writing the signs needed. Now I just followed the same process as I did on Euler’s and included a graph as well.
    My next and final Excel sheet was for Runge-Kutta, which I find fairly simple to do. I again followed the same formatting as I had previously done for the others. This time I have a k1,k2,k3, and k4. For K1 I followed the differential equation and selected the respective cells to calculate this. 2*0.2-1.7. Writing the formula for k2 was a little challenging at first because I wasn’t sure if it would work but it did. I wrote 2*(xi+h/2)-(y+h/2) while selecting the boxes corresponding to the x, y, and step size h. I did the same with k3. For k4 however it changes because now we use the whole step size h instead of dividing it by 2. My last column is yi+1 where I find the next y value. I did this by doing y+h/6*k1+2*k2+2*k3+k4. I selected each cell accordingly and got the answer. Next, I repeated the process and dragged each cell in the row down to fill the whole table.
    Overall I had no problems doing these methods in Excel. Only a missing parenthesis or typo that was not hard to fix. I liked including a graph since I wanted to see how it would look as well.
    I feel we need numerical methods included in our studies because it gives us more possibilities. There is no one right way to do things or find solutions. This broadens our minds into finding a solution another easier, faster way, rather than trying to solve something a long more complex way. This gives us the chance to also find an approximate solution in case we don’t have an actual equation to work with, only certain data. Also, if we somehow lost our computers or software this give us the knowledge find our answer by hand and therefore help us in the real world situation we are in. Which is why I’m glad it was included in our studies.
    I will be sending the Excel file and the exact solution and part III by email.

  25. Dariel Mella

    The way I did the project was using Excel which to me is by far the easiest option because you have the option to do it manually and use a bit of formulas to make it a bit easier for the lengthy process which takes for all 3 approximations. I did not get to create a calculator I only manage to obtain the basics down such as I, x, and h which are quite easy to do. The problem with the other part is that I didn’t know how to automatically update it so I had to manually update the formula which in a way is the same thing as doing it manually, so yes, the lack of knowledge on this particular software gave me a little bit of problems so it took more time to complete than I previously expected if only I can found out a way to do it automatically. Numerical methods are great for figuring out a way to get close to the answer, so it gives you options to compare different ideas to tackle a specific problem. I believe it’s a way to make excel more popular and if use it’s a whole lot faster than a calculator. I will email you the project including this word document.

  26. Earvins Baril

    The project was to test numerical methods and compare the computed values to the exact value using the exact equation. For building my numerical method calculator I used excel because I never used excel before so I thought this would be a good opportunity to learn how to use it. I encountered various challenges using excel as I did not know how to navigate and do things but once I figured it out it was simple. Using excel, I made three charts for Euler’s, Improved Euler’s, and Runge Kutta. For each method I made a chart using the method used in class. After using the formulas for the first line excel will autofill the rest of rows by using the calculation for the first row while using the numbers for the relevant rows. I didn’t know excel had this feature until I started the Improved Euler’s method, so I had done everything manually; knowing this saved a ton of time for the rest of the project.
    In comparison to the other methods learned in class, numerical methods is needed due to it’s speed. Compared to finding an exact solution to the differential equation, using numerical method is by far faster at the cost of being slightly less accurate. If only a rough estimate of the answer is needed, then using a numerical method would be the better option. Numerical methods also provide another option for those who have trouble with differential equations as it is also has a more straightforward way of computing and solving. I feel like this assignment was included in the class instead of simply using a calculator was to have us try another way of solving this. If I never had this project, I most likely wouldn’t’ve used excel or leaned how to use it. Learning this provided a fast easy way to get a close approximation to the answer.

    I will be sending parts 1-3 by email.

  27. Zijian Yu

    With the help of few of my classmates, I was able to use Excel and apply Euler’s Method, improved Euler’s Method, and the Runge-Kutta to show the slope (x, y) at each point, as well as the intermediate Y value in the improved Euler method, with the aid of a classmate and class notes. I did this by changing the values of X 0 and Y 0. To calculate k, I use excel to enter the necessary values into the equation that was given to us. Using the original formula, I did this for all Z, k, and Y (n+1) values. I had to adjust my excel formula when I got to the (k2, k3, k4) column to get a more correct value for my estimate. As a result, the K1 formula differs from the (k2, k3, k4) formula. Even though I used the same formula (k2, k3, k4), their values appeared to be different, demonstrating that outcomes can vary when performing experiments and solving everyday problems. These methods are important because scientists, engineers, and mathematicians need them in order to obtain better and more reliable results, with the goal of improving data and technology when conducting experiments.

  28. Mebi

    I utilized Visual Studio IDE to write the program in C++ language. I choose C++ because I am currently taking a C++ class, and it would be an excellent way to get more practice. I worked on the project together with one of my classmates, also experienced in C++. The process to build this project was challenging initially, but once the variables and the equations were set, everything else started falling into place. For example, once the calculator for Euler’s Method was completed, the other methods were built. One of the most challenging parts of this program was to make sure that the console displayed the output into a table grid, which took a lot of work and patience. Building the calculator was a bit challenging. It required a lot of critical thinking to set the program to focus on the function and keep the calculations at the required range.
    Moreover, I tried to keep the program as simple as possible and only use iteration to pass the value instead of increasing and moving on to the following line until the loop reached the end. This way, the computer could automatically output each result on the console. The program was set so it could work with any number of initial condition or steps and no just static solutions by selecting the console to asks the user to enter the initial values of “y,” “x,” “the value of y(x) to find it, and at last “h”; after all of the numbers are entered, the program then adds the values into the formulas already in the program and display the calculations on the console. We use numerical methods to study the approximation of errors of a numerical solution, such as a differential equation, even though we can see how some ways like the Euler’s method have a considerable margin of errors compared to the Runge-Kutta method. By doing this project, we can test the power of computer algorithms and see how it works.

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