Homework Hints Week 2: Integrating Factors – A Shortcut

Posted on February 7, 2015 by | 2 comments

Shortcuts are dangerous things – they may save you time, but they usually don’t help you understand the problem.  Because of this, it’s usually important to have a thorough grasp of the basic idea of how to solve a problem before learning the shortcut.  Since you’ve had a few days to wrestle with the “Integrating Factors” problem, I wanted to share a standard shortcut (covered in the text, but not yet discussed in class) for solving these problems, which condenses much of the algebra into two formulas.  You are welcome to use it, or not, as you prefer.

Shortcut for solving Integrating Factors problems:

Step 1:  Rewrite the differential equation in the standard form:  

\frac{dy}{dt} + p(t)y = g(t)

In practice, this usually just means getting the y and \frac{dy}{dt} on the same side, and dividing to get rid of anything in front of the \frac{dy}{dt}.

Step 2:  Find \mu, by plugging in:

\mu = e^{\int p(t) dt}

That is, integrate the function in front of y, and then raise e to the power of the result.  This gives \mu.

Step 3:  Multiply both sides of the equation by \mu, and then integrate both sides.  Notice that the integral of the left side will always equal \mu\cdot y.

Step 4: Finally, solve the resulting equation for y.  You’re done!

Special Bonus Shortcut II:  The work of Steps 3 and 4 can be condensed into the following formula, which can be used to find y directly after completing Step 2:

y = \frac{1}{\mu(t)} \int \mu(t) g(t) dt + C

That is, multiply \mu by the function g(t) from the right hand side of the differential equation, integrate, and multiply the result by $\frac{1}{\mu}$.

NOTE: The standard form mentioned in Step 1 shows up a lot – in fact, even if you are not using the shortcut formulas above, it is considered “pretty standard” to rewrite your equation in standard form before solving the problem.

An example using the Shortcut: NOTE: In the video, he uses x as the independent variable, instead of t.

 

Another example: NOTE: Towards the end of this example, when integrating the right-hand-side, he uses integral of \frac{1}{1+x^2}, which is \arctan x  — if this looks unfamiliar, you should review the derivatives of the inverse trig functions 

 

 

Happy shortcutting,
-Prof Reitz

 

 

 

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Homework Hints Week 2: Objects Falling near sea level

Posted on February 4, 2015 by | Leave a comment

Falling objects.  This video starts with some discussion of where the differential equation comes from, then carries out a problem from start to finish.  A couple of things to note:
1.  The problem uses U.S. Customary Units (feet, pounds, etc.) – in particular, acceleration due to gravity is 32 ft/sec^2  (instead of 9.8 m/sec^2).
2.  In solving the example, the differential equation is solved using the Integrating Factor method (which we will learn on Thursday).
3.  The “limiting velocity” is just another term for the terminal velocity.

 

 

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Calculus Review – some helpful hints for WeBWorK #1

Posted on February 1, 2015 by | 2 comments

EDIT 2/2/15: Added an example for finding the tangent line to a function.

Having trouble with the WeBWorK?  First, don’t panic – it’s a lot to remember!!  But do be prepared to put in some time re-learning stuff from Calculus I and II.  I’ve picked out a few video resources for you that hit some of the most important techniques  (I tried to find videos that were focussed on examples, rather than theory, since this is meant to be review).

Comments are welcome (just click the “leave a comment” button above).

  • Like a video? Leave comment and let me know.
  • Dislike a video (it wasn’t helpful/ it was confusing)? Let me know.
  • Need help with another topic (like the product rule, or equations of tangent lines, or something else)?  Let me know.
  • Have a video or other resource to suggest? Let me know!

Derivatives: Equation of the tangent line to a curve (similar to Problem 2):  This gives an example of finding the equation of a tangent line, starting with just the function and the x-value.  (NOTE: In the video, the function is an exponential function, so the numbers running around the answer all tend to have e in them – this will not be the case in the WeBWorK problem, where you will find more familiar numbers in your answer).

 

Derivatives: The Chain Rule (similar to Problem 4):  This video is short and sweet, a single example using the chain rule with a logarithmic function.

 

Integrals: U-Substitution (similar to Problems 5 & 6):

This video has three examples – the first two are most similar to what you will see in WeBWorK (the last one is a little trickier – but could be useful in the future):

 

Integration by Parts (similar to Problems 7 & 8)

This video also has a few examples – the first two will be most useful for the WeBWorK:

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Getting Started with WeBWorK

Posted on January 28, 2015 by | Leave a comment

WeBWorK is accessible from on and off campus, anywhere you have access to the internet.  Your first WeBWorK assignment, a review of important Calculus skills, is due next Tuesday, February  3th, at midnight. 

To get started, you must complete the following three steps.

Step 1.  Log in to WeBWorK here:  http://mathww.citytech.cuny.edu/webwork2/MAT2680-S15-Reitz/.  I have created Usernames and Passwords for each student registered for my class.

Username.  Your username for WeBWorK consists of your first initial plus your last name, all lowercase (for example, John Smith would have username ‘jsmith’).

Password.  Your temporary password is the same as your username (if your username is ‘jsmith’, your password is currently ‘jsmith’).

Step 2.  Change your password and update your email address.  To do this, select “Password/Email” from the main menu on the left.  Use whatever email address you like (I suggest using one that you check often).

Step 3.  Complete the first assignment, titled CalculusReview, by clicking on it on the main screen.

If you have any trouble – either with logging in, or with completing the assignment, post a comment here or send me an email and I will get back to you.

WeBWorK Tips:

  1. Click on a problem to see the details (the list of problems appears in the menu on the left).  Enter an answer and hit “Submit Answers”.  Don’t worry, if you get it wrong you can try it again.
  2. You can work on the problems in any order you wish.  You can do some problems now, and come back and do the rest another day (your work will be saved, as long as you submit your answers).
  3. If you want to print out a copy of the assignment, click on the assignment name in the main menu on the left, and then click the link in the main screen area that reads “Download a hardcopy of this homework set.”

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Welcome and Getting Started

Posted on January 27, 2015 by | Leave a comment

This course is MAT 2680, Differential Equations, taking place in the Spring 2015 semester with Professor Reitz.  We will be using this website in a variety of ways this semester – as a central location for information about the course (assignments, review sheets, policies, and so on), a place to ask and answer questions, to post examples of our work, and to talk about  mathematics, physics, reality and so on.

Getting Started

Anyone on the internet can look around the site and see what we are doing, and even leave a comment on one of the pages.  However, only registered users can create new posts and participate in the discussion boards.

How do I register?

You will need to do two things:

  1. If you have not used the openlab before, you must first create an account.  You will need access to your citytech email address  for this.  Detailed instructions for signing up on the OpenLab can be found here.
  2. Once you have created an account on the OpenLab, log in and then join this particular course, 2015 Spring – MAT 2680 Differential Equations – Reitz.  To do this, first click the “Course Profile” link at the top left of this page (just above the picture).  Then click the “Join Now” button, which should appear just underneath the circular picture filled with green words.

Problems with the OpenLab or with your CityTech email:

Please let me know if you run into any problems registering or joining our course (send me an email, jreitz@citytech.cuny.edu).  I also wanted to give you two resources to help out in the process:

1.  For problems with your citytech email account, contact the Student Computing Helpdesk, either in person, by phone, or by email:

Student Computing Helpdesk
Location: Namm First Floor – Information Booth
Hours: Monday, Wednesday & Friday: 9:00am – 5:00pm
Tuesday & Thursday: 9:00am – 6:00pm
Phone: 718.260.4900
E-mail: Studenthelpdesk@citytech.cuny.edu

2. For problems registering for the OpenLab, contact the OpenLab admin team, either by email at openlab@citytech.cuny.edu, or by following this link.

 

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