Category Archives: Resources

Getting Started, How-Tos, and so on

Final Exam Review Sheet is posted

UPDATE 5/17/14:  Corrected ALL answers to problem #17 (numerical methods).

Posted on the “Exam Reviews” page.  As always, let me know if you find errors or have questions.

Best,

Prof. Reitz

Exam 3 Review Sheet is posted

UPDATE 5/10/2014: Correction to answer #3 — it should be “-” not “+”.  (This also affects the value y(1.2)).

UPDATE 5/8/2014: Runge-Kutta data & formulas have been added.

UPDATE 5/6/2014, 1:20pm: Correction to answer #4d  —

4d:  $49e^{16t}+cos{7t}$

It can be found on the “Exam Reviews” page.

NOTE: some of the information about the Runge-Kutta method is missing – this will be added within the next few days.

Best of luck,
Prof. Reitz

MAT3880: Partial Differential Equations through Mathematical Models in Biology

Hi everyone,
If you’re interested in pursuing differential equations further, I highly recommend this course being taught in the fall.
Best,
Prof. Reitz
—————————————-
Attention Differential Equations Students!
One of the most fascinating and important areas of active mathematical research is Mathematical Biology—the art and science of studying life through mathematics and the mathematics of life.  MAT3880: Partial Differential Equations through Mathematical Models in Biology covers important current topics in this important field.  The course provides a solid foundation in the concepts and methods of partial differential equations within the context of biology
The class meets MW 2:00-3:15pm.

WeBWorK error: Assignment 9, problem 4 parts (e) and (g)

Hi everyone,

There is an error in Assignment 9, problem 4, parts (e) and (g) – several students have reported entering the correct answer (which I have confirmed), but WeBWorK rejects it.

Even more strange, part f is graded correctly, even though it includes the answer that was rejected in part e.

I am following up with our WeBWorK administrator on this problem.  For now, please complete all parts of the problem except (e) and (g).

Sincere apologies for any mental or emotional harm this may have caused.

Regards,
Prof. Reitz

Exam #2 Review Sheet is posted

NOTE: Problem #11 was updated on 3/24/14.

You can view it by clicking here.  Note: the answer key will be completed this week.

More on nonhomogeneous equations and WeBWorK 9

Or:  What to do when everything cancels out, and I’m left with nothing….

I wanted to address a problem that shows up when solving nonhomogeneous equations.  Consider this problem, similar to those in WeBWorK 9:
$y'' + 8y' +15y=2e^{-3t}$

A good first guess for a solution would be: $Y(t) = Ae^{-3t}$.  However, there is a problem with this – when you try to find the constant $A$, everything cancels out on the left side and you end up with something like:  $0 = 2e^{-3t}$.

Why is this?  It’s exactly because your  guess, $Y(t) = Ae^{-3t}$happens to be a solution to the homogeneous equation:
$y'' + 8y' +15y=0$

When this occurs, you need to adjust your guess – we do that by multiplying by $t$, so a correct guess in this case should be:
$Y(t) = Ate^{-3t}$

If this guess also fails to work (as might happen in the case of a repeated root for the homogeneous equation), multiply by $t$ again:  $Y(t) = At^2e^{-3t}$

Let me know if you have any questions,
Prof. Reitz

A few notes on WeBWorK 5b – Euler’s Method

Hi everyone,

I just wanted to send a few comments about WeBWorK assignment 5b.  The problem consists of three parts – in the first part, we use Euler’s method to generate a sequence of approximate values of y for various values of t (in my version of the problem it’s t=0, t=0.2, t=0.4, t=0.6, t=0.8, and t=1). Of course, these values are not exactly correct – they are approximations.

In parts b and c of the problem, we compare the approximate values with the actual values – this will give you some idea of how effective Euler’s method is.  In part b, we find the actual solution to the differential equation (using the techniques from Exam #1).  In part c, we compare (for various values of t), the actual value, given by the solution we found in part b, and the approximate value found in part a. To compare, we subtract the approximate value from the actual value (and then take the absolute value, since we don’t care about positive or negative, we only care about how far apart they are).

One more comment about notation:  the symbol y with a dot over it means the same thing as y’, or dy/dt – that is, it means the derivative of y.  This is fairly common notation, but we have not previously run into it this semester.

I hope this helps clear up confusion about parts b and c – let me know if you have questions.

Regards,
Prof. Reitz

Exam #1 Review Sheet UPDATES

Hi everyone,

I’ll post updates & corrections to the review sheet here – check back often for the latest.

• UPDATE #1:  Corrected answer to problem 1 (exponent should have -1/30t, instead of 1/30t)
• UPDATE #2: Corrected answer to problem 4 (exponent in the denominator should be 14, not 13)
• UPDATE #3: Corrected answer to problem 8 (things were off – something was clearly wrong with the left hand side)

Integrating Factors: A Shortcut

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 week or so 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:  Find $y$, by plugging in:

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

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.

Happy shortcutting,
-Prof Reitz

Tutoring for MAT 2680

The Math Department provides free tutoring by Math Specialists four days a week. They are (in the words of my colleague) “experts at Differential Equations”.  This is a fantastic resource.  I’ll paste the schedule below – see the attached flyer for more details.

SCHEDULE:
MONDAYS & WEDNESDAYS 1:00PM-6:00PM IN N604
THURSDAYS 3:00PM TO 6:00PM IN N604
FRIDAYS 10:00AM-4:00 PM IN N723

Spring 2014 Math Tutoring flyer