Author Archives: Jonas Reitz

OpenLab #2: Differential Equations Study Guide (part 1 – topic selection)

Posted on March 18, 2015 | 1 comment

Hi everyone,

This is the beginning of a multi-part OpenLab assignment focussed on building resources for you and your fellow students.  Our ultimate goal will be to create a study guide for this course, with videos and other resources for each topic.

Assignment (Due next Thursday, 3/26).  Your first assignment is to choose a topic to work on.  Fill in the form below, selecting THREE different topics that you would be interested in researching for our Study Guide (you will be assigned one of your three choices).  Feel free to choose a topic that we have not yet studied (you will be given ample time to work on your portion of the Study Guide after we cover the topic in class).

You must be logged in to the OpenLab to complete this form. Having trouble with your account? Look here for help and resources: https://openlab.citytech.cuny.edu/2015-spring-mat-2680-reitz/?p=39

 

 

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Office hours cancelled today (Thursday, 3/19)

Posted on March 18, 2015 | Leave a comment

Hi everyone,

My office hours are cancelled today (Thursday, 3/19) as I will be observing another class.  I will be available directly after class for brief questions – if you want to meet with me for a longer period please let me know.

Best,
Prof. Reitz

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Second order ordinary homogeneous differential equations with constant coefficients

Posted on March 17, 2015 | Leave a comment

Here are some video resources showing complete examples (from Khan Academy).  I hope they help!

-Prof. Reitz

Example 1 — in which the characteristic equation has two distinct real roots.

 

Example 2 — in which the characteristic equation has one repeated (real) root.

 

Example 3 — in which the characteristic equation has complex roots.

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Euler’s Method Exercises

Posted on March 13, 2015 | 1 comment

Hi everyone,

Here are some examples of Euler’s Method problems for you to try.  These are for practice only, they will not be collected.
NOTE: The first example is the one we solved in class on Thursday – I’m including it here for your reference.

Questions? Problems? Leave a comment on this post and I’ll respond here.

Example 1.  Given the initial value problem \frac{dy}{dt} = 3-2t-0.5y with y(0)=1, approximate the value of y(1).  Use 5 steps.

Example 2.  Consider y' = y \sin{t} -1.  Suppose y(0)=2.  What is y(3)?  Use a step size of h=0.5.

Example 3. If y'=\frac{x-4}{x-y+1}, use Euler’s method with 4 steps to approximate y(8) for the particular solution satisfying y(3)=3

Example 4.  Given \frac{dy}{dt}=e^\frac{t^2}{y^2+1} - y with y(-0.82)=0.97, estimate the value of y(1.28) using 6 steps.

Solutions to these examples are provided here for your reference – if you notice an error, please let me know by leaving a comment below (or sending an email).

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Office Hours rescheduled Thursday 3/11/15

Posted on March 11, 2015 | Leave a comment

Hi everyone,

I will be observing another class from 9-10am this Thursday 3/11, so my office hours will take place:
12:30pm – 1:30pm, on Thursday 3/11, in N707

Best regards,
Prof. Reitz

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Office Hours rescheduled today (2/26)

Posted on February 25, 2015 | Leave a comment

Hi everyone,

I was running a workshop from 8:30 – 10:00am today (Thursday, 2/26), so my office hours will take place directly after class, 11:15-12:15, instead of before class.

Sorry for any inconvenience,

Prof. Reitz

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Exam 1 Review is posted

Posted on February 21, 2015 | 2 comments

Hi everyone,

Exam #1 will take place on Thursday, March 5th, during class.  The review sheet for Exam #1 has been posted on the “Handouts” page.  Let me know if you have any questions.

Regards,
Prof. Reitz

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WeBWorK – how to enter infinity

Posted on February 17, 2015 | 1 comment

Hi everyone,

It seems that entering “infinity” in WeBWorK can vary from problem to problem – but the most common versions are:

infty, -infty

infinity, -infinity

Inf, -Inf

I think the problems in this set prefer “infinity” and “-infinity”.

I was unaware of these variations – my apologies!  Let me know if you are still having trouble.

Regards,
Prof. Reitz

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OpenLab #1: Advice from the Past

Posted on February 10, 2015 | 67 comments

Last Spring I taught this same course for the first time.   At the end of the semester, I gave my students the following assignment:

Imagine that you are invited to speak on the first day of MAT 2680, to give advice to entering students. Write at least three sentences … describing what you would tell them.

To see the assignment and the students’ responses, follow this link.

Your assignment, due on Thursday 2/19 at the start of class, is to:

  1. Read through ALL the responses (there are 53 of them – many are quite short).
  2. Write a reply to this post (150 words minimum, not including quotations) responding to all of the following:
    1. What advice seemed most relevant to you personally? Why? (you can copy/paste a short statement, or put it in your own words)
    2. Based on this advice, what changes can you make right now to help you with this course?

Extra Credit. For extra credit, write a response to one of your classmates’ comments.  Do you have any advice?  Be kind.

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Homework Hints Week 2: Integrating Factors – A Shortcut

Posted on February 7, 2015 | 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.

https://www.youtube.com/watch?v=Et4Y41ZNyao

 

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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