PHYSICS 312
Problem set 6

Given March 24 1999
Due April 7 1999
Problem 1:
Using the infinite series representation for the Bessel function verify

$$\frac{d}{dx}(x^{-n}J_n(x))=-x^{-n}J_{n+1}(x))$$

$$\frac{d}{dx}(x^nJ_n(x))=-x^nJ_{n-1}(x))$$

(The last of of these formulase were proven in lecture 26.
Problem 2:
Show that the function

$$\phi(x,y)=\sin(\pi x)\sin(2\pi y)-\sin(2\pi x)\sin(\pi y)$$

is an eigenfunction of the triangle T bounded by the lines y=0,y=x,x=1 associated with the differential equation

$$\nabla^2\phi=-\lambda^2\phi$$

and the boundary condition

$$\phi=0$$

on the boundary of T. What is the eigenvalue $\lambda^2$ asociated with $\phi$?
(This problem was given again as problem 4 of midterm review problem set 2001.) Problem 3:
Show that

$$u(r,t)=t^{-3/2}\exp(-\frac{r^2}{4t})$$

is a solution to the three-dimensional heat equation

$$\nabla^2u=\frac{\partial u}{\partial t}$$

in spherical coordinates.
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