APM346-2018S > APM346––Home Assignments

An ODE

(1/2) > >>

Jingxuan Zhang:
I am referring to Q2.3 on
http://www.math.toronto.edu/courses/apm346h1/20181/PDE-textbook/Chapter8/S8.P.html

So how do I actually solve
$$\sin^2\phi \Phi''+\sin\phi\cos\phi \Phi' -(l(l+1)\sin^2\phi-m^2)\Phi=0$$
which, suppose it's correctly derived, has cost me an entire afternoon? I remember I certain remark
in lecture that there should be constrain on $m$ and some thing like $(x+iy)^m$, but that part of my note
is very much blurred.

I heuristically plugged in $\sin,\cos,\sin^2,\cos^2,\sin\cos$ but there does not seem to be a good cancellation.

Victor Ivrii:
$|m|\le l$ and both are integers

Jingxuan Zhang:
But what then is meant by $(x+iy)^m$? are these the solutions? apparently it doesn't seem to be.

Victor Ivrii:
In case of 2 variables $(x\pm iy)^{|m|}$ are solutions (harmonic functions)

Jingxuan Zhang:
But in the hint you say the solution should be in the form of trig polyn? and the ODE should have one variable? how am I supposed to interpret this $(x\pm iy)^{|m|}$?