Homework O: Due Monday September 11th
Problem 1. Solve the following differential equations:
(a)
(b)
dy
dx
dy
dx
+ x2 y = x2 with y(0) = 1.
= x2 ey with y(1) = 1. Does the solution approach infinity at a finite value of x?
Problem 2. Find the explicit solution to the boundary value problem
x2 y 00 − xy 0 + 4y = 0 ,
y(1) = 0 ,
y(2) = 1 .
Problem 3. Calculate the transient solution and the steady-state solution for
y 00 + y 0 + y = sin(ωt) ,
y(0) = 1 ,
y 0 (0) = 0 ,
where > 0 is constant. Write the steady-state solution in the form y = A cos(ωt − φ) for
some A and φ depending on ω. Plot A versus ω for a fixed small.
Problem 4. Calculate the eigenvalues and eigenfunctions for the following Sturm-Liouville
problem for y(x):
y 00 + λy = 0 ,
0 < x < L,
with the either of the three sets of boundary conditions:
(a) Absorbing Boundary Conditions: y(0) = y(L) = 0,
(b) No-flux Boundary Conditions: y 0 (0) = y 0 (L) = 0,
(c) Periodic Boundary Conditions: y(0) = y(L) and y 0 (0) = y 0 (L) .
Problem 5. Calculate the solution for x = x(t) and y = y(t) to
dy
= −y ,
dt
dx
= y + y 2x ,
dt
with y(0) = 1 and x(0) = 2. In addition, determine y as a function of x by eliminating t.