Final Exam

Differential Equations - Spring 2009 - Marlboro College

Please do read the instructions first.  Good luck.  - Jim M

instructions

Your mission is to use the six differential equations below 
to showcase your understanding of the material we've covered this term, namely
	* exact solutions, including separation of variables, integrating factors, and so on,
	* changing the physical dimensions to find a convenient form of the equation,
	* graphical approaches, including direction fields and solution curves, and phase space
	* numerical methods such as runge-kutta and heun,
	* initial value problems and integration constants
	* system of first order equations,
	* the use of eigenvalues and eigenvectors,
	* critical points and the qualitative long-term behavior of the solutions,
	* stability and/or chaos in particular circumstances.

All greek letters (e.g. α,β,γ,Λ,...) are constants, 
while roman letters (e.g. x, y, z, u, v, t, ...) are variables.

While you shouldn't try to include all of these techniques in every equation,
most of them will be appropriate for at least one of the equations.  And many
of the equations will naturally use several of these ideas.  I'm expecting exact solutions 
for some, graphs of diverse sorts for several, numerical solutions from a 
particular initial conditions for others, and so on.  You're welcome to use more 
than one approach on the same problem, for example adding a numeric solution to one 
that you can do analytically, as a way to look at the accuracy of the numerical method.

In each case I've simply given an equation, and not specified a particular initial condition;
however, I encourage you to choose an initial condition (or several) and work from there.
You may also want to choose specific values for any constants - or even turn numbers
like 4 into constants like β and discuss what happens for similar equations.  Whatever
you do, be clear about your choices and assumptions.

You have one day to do the exam, which is due by email, posted on the assigments 
page, and/or tucked under my office door by midnight Friday May 8.  You're welcome 
to use any format you like: Mathematica, Excel, python, by hand, or whatever.

Sources such as our text or the 'net are OK; however, if so you must cite them.
Remember that you're being evaluated on your understanding of the material,
as I can see it in what you write - not just "the right answer" - so be clear in 
your presentation.  

Asking other people for help is not allowed.

If you have questions, you can email me or phone at either 247-0857 (home) or 258-9255 (office).

1.

((dy)/(dx))+(e^(2x))+y-1=0

2.

((dy)/(dx))+α (x^2)y=0

3.

(((d^2)y)/(d(x^2)))-cos(x)-((dy)/(dx))-y=0

4.

(((d^2)y)/(d(x^2)))-(1-(y^2))((dy)/(dx))+y=0

5.

((dx)/(dt))+x+y+1=0
((dy)/(dt))-2x+y-5=0

6.

((dx)/(dt))-(1-(((x+y))/2)) x=0
((dy)/(dt))-(2x-1) (y/4)=0