HWK 1, Due Thur Feb 10
  1. Solve x2 y'' + x y' -y = f(x) with y(0)=y(L)=0 by variation of parameters as well as by using a Green's function . Find the explicit solution in the special case when f(x)=x. [references: Wylie & Barett 2.12, p. 173, but W&B avoid mentioning Dirac delta functions; Greenberg's "foundations" 22.5 p.434 uses Dirac deltas, it is also a more general treatment and shows how to deal with non-homogeneous boundary conditions]
    As discussed in class x=0 is a regular singular point for this ODE. Are we free to impose any boundary condition at x=0? In other words, how much control do we have over y(0) ?

  2. Solve, sketch and discuss as a function of the undetermined constant of integration. What are the possible y(0) values?
    1. y'+y/x = 1
    2. y'+y/x2 = A (sketch for A=0 only)

  3. Use Euler's method to solve
    1. y'= y, y(0)=1. Make a plot of error at time 1 vs. time step
    2. y'=t2 - y2, y(0)>0. Use a time step of 1, 1/2, 1/4, 1/8,... Try to determine the long time behavior of y(t) for various initial conditions. Discuss and try to explain whatever curious features may show up.