Math 425 / AMCS 525  Notes and Homework
Thursday January 14, 2016
Topics for this week 
 Review of ODEs, with an eye toward
PDEs
Examples 
 Linear and separable ODEs, solve
y' + xy = x two
ways; solve y" + 5y' + 4 = 0.
 Explain why there is only one solution to an initialvalue problem
for the above equations.
 Recall basic theorems of vector calculus (Green, Gauss, Stokes).
 Fun with integration by parts.
First Homework Assignment  due Tuesday, January 19
 Reading: Read
these
notes on ODEs

 Find the general solution of x'(t) +
x^{2}sin(t) = 0.
 Solve the initialvalue problem: x'(t) +
x(t)cos(t) = 0, x(π)=100
 Find the general solution: 2y'' + 5y' + 2y = 0.
 Find the solution of the initialvalue problem:
5y'' + 8y' + 5y = 0, y(0) = 1,
y'(0)=0.
 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.
 Prove that the solution of the initialvalue 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).
 (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?
 For a function u(x,y) of two variables, its
Laplacian is defined to be
Δu = u_{xx} + u_{yy}.
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 u_{xx} + u_{yy} = 0) ?