The book sections covered on the final will be a subset of those covered in the homework, the last one of which was due Monday.
In particular, there is no homework nor will there be final exam questions about Laplace transforms nor series solutions to differential equations.
Here are some questions on most of the topics on the final to get you started.
1. Let A be a square matrix with A*A = Identity. Let M be a square matrix, such that AMA = M. Let v be an eigenvector of M with eigenvalue e.
a) Show that Av is one too.
b) Assuming that, show that v+Av, v-Av are too.
2. Find the exponential of t times the matrix M
[2 0 3 9 0]
[0 2 1 6 0]
[0 0 2 0 0]
[0 0 0 2 0]
[0 0 0 0 3]
Hint: it would be easier if you subtracted 2 * Identity... how do you make use of that?
Solve the ODE system y' = My for arbitrary initial conditions.
3. Let y' = f(t,y).
a) What is the general form of f, if this equation is separable and linear homogeneous?
b) What is the general form of f, if this equation is separable and linear but not necessarily homogeneous?
4. Let c_0 = 0, c_{n+1} = c_n + 2n + 1. Find c_0,c_1,c_2,c_3,c_4,c_5 and guess a general formula for c_n. Then determine the radius of convergence of the power series \sum_n c_n x^n.
5. Assume that
f(x) y dx + (x+y) dy = 0
is exact. Determine the possibilities for f, and solve this ODE. (You should have two constants in your answer: one because you don't know f precisely, and the other from solving the ODE.)
