Homework #4

Due Friday of Week 4 at 5pm.

1. Consider z(t) = √3cos(ωt) + sin(ωt). Write z in the form Acos(ωt+δ). Chose a solution method that does not rely on trig identities. It is more elegant and instructive to use complex numbers.

2. Complete the finite line charge problem that we started in class (see notes from Day 6). i.e. Find an expression for potential as a function of position. When evaluating the integral, first make the integral dimensionless (extract all the physics), then look up the integral in a table.

3. Find the general solutions for the following ODEs of order n = 2 (general solution will contain n arbitrary constants and n linearly independent functions). All constants and unknown functions are real-valued, however, it may be easiest to find complex-valued solutions first, and then translate back to the real-world.

a) y’’ + y = 0

b) y’’ + y’ + y/2 = 0

c) y’’ + 6y’ + 9y = 0

4. a) In class we found a two Bessel functions by applying the Frobenius method to the Bessel equation. Calculate the first and second derivatives of these Bessel functions and confirm that they satisfy the appropriate Bessel equation.

b) Do the same thing for a Legendre polynomial (chose a Legendre polynomial that is more interesting that y = 1!)

5. Use the Frobenius Method to solve the following ODEs:

a) zy’’ - 2y’ + 9(z^5)y = 0

b) z²y’’ - (3/2)zy’ + (1 + z)y = 0