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Exact Equations

Posted on May 1, 2015 by | 1 comment

An exact equation is a first order differential equation. The steps to solving an exact equation is to check if MY  is equal to Nx. to check if they are equal you have to derive by M by y and derive N by x. the next step is to Integrate M with respect to X and obtain  and have to make sure to add the h(y) or else you will get an error later on. next take the derivative of  what the step done before but with respect to y.next you have to it equal to N and make sure that h(y) was also derived in the previous step. next solve for h'(y). after solving h'(y) we now integrate h'(y) to find h(y). After this we plug in h(y) to to the step where we integrated M with respect to x and make sure the equation is equal to C.

Next I will show an example for Exact equations:

finding the general solution for 2xy-9x^2+(2y+x^2+1) \frac{dy}{dx}=0.

the first step is to derive 2xy-9x^2 with respect y and 2y+x^2+1 with respect to x.

For 2xy-9x^2 we get 2x and for 2y+x^2+1 we get 2x as well so they are both equal to each other.

the next step is to integrate M with respect to y.

\int 2xy-9x^2 dy.

after integrating you should have gotten x^2y-3x^3+h(y).

the next step is deriving what you just integrated with respect to y.

when deriving you should get x^2+h'(y).

next make x^2+h'(y) equal to N which is 2y+x^2+1.

x^2+h'(y)=2y+x^2+1

the next step would be to find h'(y).

h'(y) would be 2y+1.

next we integrate h'(y).

\int h'(y)= \int 2y+1

after integrating we get

h(y)=y^2+y.

after finding h(y) we plug it into the equation we got when we integrated M with respect to y.

x^2y-3x^3+h(y)

the equation should now be x^2y-3x^3+y^2+y=c.

here are  videos on how to do exact equations if you need a better understanding:

https://www.khanacademy.org/math/differential-equations/first-order-differential-equations/exact-equations/v/exact-equations-example-1

Resources:

http://tutorial.math.lamar.edu/Classes/DE/Exact.aspx

class notes

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Complex Roots(of the Characteristic Equation)

Posted on April 26, 2015 by | Leave a comment

Complex Roots relate to the topic of Second order Linear Homogeneous equations with constant coefficients. The Second Order linear refers to the equation having the setup formula of y”+p(t)y’ + q(t)y = g(t). Constant coefficients  are the values in front of the derivatives of y and y itself. Homogeneous means the equation is equal to zero.So a homogeneous equation would look like

y”+by’ + cy = 0 or y”+p(t)y’ + q(t)y = 0.

When solving the general solution of the homogeneous equation, we have to look for the characteristic equation which is r^2 + br + c = 0. But how would you get there? To start we are given the differential equation of

y = e^rt

which can be used to look for the characteristic equation. To start we look for the derivative of the differential equation then replace y and its derivatives with what we found which the homogeneous equation would look like

(r^2)e^rt + bre^rt + ce^rt = 0

You can see there is a common value which is e^rt you can factor out. Once you factor it out, you can divide e^rt and this is the characteristic equation,

r^2 + br + c = 0

The general solution of the homogeneous equation will have 3 different results depending on the roots which are found from the characteristic equation. This is where Complex Roots come into play. Complex Roots is when the roots have an imaginary number or i which looks like r =  a ± i*b. The general solution will be

 y = C1e^(a + ib)t + C2e^(a – ib)t

But there is a problem when you want to solve for a particular solution how would you get rid of the i? So the general solution is still not done. With the general solution, y = C1e^(a + ib)t + C2e^(a – ib)t you can right away separate the roots which will become y = C1e^at * e^(ibt) + C2e^at * e^( –ibt). Next you factor out e^a which makes the equation y = e^at(C1e^(ibt) + C2e^(-ibt). This is where the Euler’s Formula is introduced which says that

e^it = cos(t) + isin(t)

So the equation is written as such

y = e^at(C1(cos(bt) + isin(bt)) + C2(cos(-bt) + isin(-bt)))

This is where your knowledge of trig identities comes to play. Two trig identities which can be used are

sin(-x) = -sin(x), cos(-x) = -cos(x)

This would replace the negative values in the sin and cos but we are left with the complex part i. To get rid of the complex part, the equation is rearranged so that the constants can be created into a new constant that will work with real and imaginary numbers.

y = e^at((C1 + C2)cos(bt) + (C1i – C2i)isin(bt)))

y = e^at(Acos(bt) + Bsin(bt)), general solution

Sample Problem:                                                                                                                                y” + 4y’ + 5y = 0, y(0) = 1, y'(0) = 0                                                                                      y=e^(rt), y’ = re^(rt), y” = (r^2)e^(rt)                                                                                     e^(rt)*(r^2 + 4r +5) = 0

Since the roots are the easy to separate, we use the quadratic formula to look for the roots which you will end up with                                                                                            r = -2±i

The General Solution will equal to

y = e^(-2t)*(Acos(t) + Bsin(t))

For the particular solution, we solve for A and B using y(0) = 1 and y'(0) = 0.   y(0) = 1                                                                                                                                                      1 = e^(-2(0))*(Acos(0) + Bsin(0))                                                                                                1 = A

y'(0) = 0                                                                                                                                                  y’ = -2Ae^(-2t)cos(t) – Ae^(-2t)sin(t) – 2Be^(-2t)sin(t) + Be^(-2t)cos(t)                    0 = -2(1)e^(-2(0))cos(0) – (1)e^(-2(0))sin(0) – 2Be^(-2(0))sin(0) + Be^(-2(0))cos(0)                                                                                                                                 0 = -2 + B                                                                                                                                                B = 2

The Particular Solution:

y = e^(-2t)*(cos(t) + 2sin(t))

Videos to help with Studies:

Complex Roots Part 1

Complex Roots Part 2

Complex Roots Part 3

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OpenLab #4: Study Guide

Posted on April 14, 2015 by | 7 comments

Hi everyone,

This post contains the instructions for the Study Guide project.  Our goal is to create a study guide that will be useful both for the students in this class, and for future students.  This project will require an investment of time & energy on your part, and will be given a corresponding weight in your OpenLab grade (equivalent to two OpenLab assignments).

NOTE: your Study Guide should NOT be a comment in reply to this post, but instead should be a new Post that you create (see the “Resources” section below for help on this).  Creating a post means that you can save your work, and come back and edit it later.  You also have much more control over things like formatting, inserting images and videos, and so on.

Instructions.

  1. To get started, create a new post (HINT: click the “plus” sign in the grey bar at the top of the screen).
  2. Give your post a title that makes sense
  3. Select the Category “Study Guide” (from the box on the right side)
  4. Add at least 3 tags (you choose) (from the box on the right side, below “Categories”)
  5. Your post should include
    1. Overview.  Explain what the topic is about.
    2. Sample problem.  Give a sample problem, and explain how to solve it.
    3. Videos or images.  Please include at least 3 videos or images in your post that you personally found helpful.  Be sure to include a brief description.
  6. 500 words minimum (no maximum – write as much as is needed).
  7. Don’t forget to click either “Save Draft” or “Publish” from the topmost box on the right side.  “Publish” will make your post public on the main page of the site (but you can still edit it later), “Save Draft” will save your work without making it public.

Question: What if more than one person has the same topic?

  • Option 1: you can work together on a single post (this is preferred – but you will each be responsible for writing at least 500 words)
  • Option 2: you can work independently (you each create your own post)

Question: When is it due?

  • For topics covered BEFORE Spring Break (through Section 3.7), your Study Guide is due April 30th.
  • For topics covered AFTER Spring Break, your Study Guide is due May 18th (beginning of Finals Week).

Question: I saved a draft, how can I find it so I can work on it some more?

  • Saved drafts of posts can be accessed through the Dashboard:
    • Make sure you are logged into the OpenLab
    • To access the Dashboard, click the link “MAT 2680 Differential Equations” in the gray bar at the very top of the screen
    • Select “Posts” from the menu on the left side of the screen, and scroll through to find your post.  If you wish, you can show only posts that are “Drafts” by clicking the appropriate link near the top of the page (below the main title “Posts”)

Resources that may help you:

  • How to create a new post.
  • How to insert a video or image – look at the help topics here.
  • How to use mathematical notation on the OpenLab (like \int \frac{e^t}{t^2} dt).  This is done using the (very common in professional circles) mathematical markup language LaTeX — if you ever publish professionally in math or most sciences, you will end up learning this!  Luckily, it’s not too hard to get started — take a look at this assignment from a past Calculus class, which includes an introduction to writing LaTeX on the OpenLab.

Extra Credit

For extra credit, comment on another student’s Study Guide post. Your comment must say something substantive (more than just “great post!” or “very helpful” or “I don’t understand”) – tell us why you feel that way, or add something to the post that you think is missing, or (politely) point out an error if you think you see one, or suggest another video or other resource etc.  You can earn up to 3  extra credit points by commenting on multiple posts.

 

Study Guide Topic Assignments

If you do NOT appear on this list, please contact me immediately!

Study Guide Topic Username
2.1 Linear Equations; Method of Integrating Factors rahmanhasan718
2.1 Linear Equations; Method of Integrating Factors rmorel91
2.2 Separable Equations dhiraj
2.2 Separable Equations jiwei
2.2 Separable Equations (Homogeneous) diallo11368
2.2 Separable Equations (Homogeneous) ricardoferro
2.4 Bernouli Equations mrknowit22
2.4 Bernouli Equations wenyuli
2.4 Difference between Linear and Nonlinear Equations (Existence and Uniqueness) attareb212
2.6 Exact Equations vliang88
2.6 Exact Equations kmendez1994
2.7 Numerical Approximations: Euler’s Method samsonx711
2.7 Numerical Approximations: Euler’s Method dsantos
3.1 Homogeneous Equations with Constant Coefficients (second order linear) mattiie
3.1 Homogeneous Equations with Constant Coefficients (second order linear) jdelgado
3.1 Homogeneous Equations with Constant Coefficients (second order linear) enriquebron6
3.3 Complex Roots (of the Characteristic Equation) chand
3.3 Complex Roots (of the Characteristic Equation) rex19941
3.3 Complex Roots (of the Characteristic Equation) ltsakuxsasu648
3.4 Repeated Roots steven328
3.4 Repeated Roots hxie
3.4 Repeated Roots carolinam926
3.5 Non-homogeneous Equations; Method of Undetermined Coefficients pak1s0ul
3.5 Non-homogeneous Equations; Method of Undetermined Coefficients chowdhuryshawn
3.5 Non-homogeneous Equations; Method of Undetermined Coefficients rana
3.7 Electrical Circuits skane17
3.7 Electrical Circuits anzamul hyder
5.2 Series Solutions Near an Ordinary Point kumar
5.2 Series Solutions Near an Ordinary Point jramroop4
6.1 Definition of the Laplace Transform christianpinto
6.1 Definition of the Laplace Transform abraham
6.2 Laplace Transform: Solution of the Initial Value Problems (Inverse Transform) vanessa2793
6.2 Laplace Transform: Solution of the Initial Value Problems (Inverse Transform) aayush
8.1 The Euler or Tangent Line Method medinalex13
8.2 Improvements on the Euler Method danielmwong
8.2 Improvements on the Euler Method 520crystal

 

Grading: 10 points total (equiv. to 2 OpenLab assignments)

  1. Basics (one point each, 4 total):
    1. Title that makes sense
    2. Category “Study Guide”
    3. At least 3 on-topic tags
    4. 500 words minimum
  2. Content (two points each, 6 total):
    1. Overview
    2. Sample problem/solution
    3. Three videos/images with brief description

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