Math 425 / AMCS 525 - Notes and Homework Thursday January 14, 2016

Topics for this week -

1. Review of ODEs, with an eye toward PDEs

Examples -

1. Linear and separable ODEs, solve y' + xy = x two ways; solve y" + 5y' + 4 = 0.
2. Explain why there is only one solution to an initial-value problem for the above equations.
3. Recall basic theorems of vector calculus (Green, Gauss, Stokes).
4. Fun with integration by parts.

First Homework Assignment - due Tuesday, January 19

1. Find the general solution of x'(t) + x2sin(t) = 0.

2. Solve the initial-value problem: x'(t) + x(t)cos(t) = 0, x(π)=100

3. Find the general solution: 2y'' + 5y' + 2y = 0.

4. Find the solution of the initial-value problem: 5y'' + 8y' + 5y = 0, y(0) = 1, y'(0)=0.

5. Solve the following system of differential equations for x(t) and y(t):
x'(t) = x(t) - 4y(t) , y'(t) = x(t) + y(t), subject to the initial conditions x(0) = 1 and y(0) = 1.

6. Prove that the solution of the initial-value problem u'' + cu = 0, u(0) = a, u'(0) = b for c < 0 exists (easy - just write it down) and is unique (to do this, "factor" the operator and then apply the theorem on page 4 of the notes twice).

7. (a) Torricelli's law states that fluid will leak out of a small hole at the base of a container at a rate proportional to the square root of the height of the fluid's surface from the base. Suppose that a cylindrical container is initially filled to a depth of one foot. If it takes one minute for three quarters of the fluid to leak out, how long will it take for all of the fluid to leak out?

(b) It is desired to design a "water clock" by making a container that is in the shape of some surface of revolution with a small hole in the bottom, so that as the water empties out of the hole, the water level in the container falls at a constant rate. What should be the shape of the container?

8. For a function u(x,y) of two variables, its Laplacian is defined to be Δu = uxx + uyy. Which radial functions (i.e., functions of the polar coordinate r but independent of θ) are harmonic (i.e., satisfy the PDE Δu = 0, or uxx + uyy = 0) ?