HA 10, trouble with question 2

[Shaghayegh A]

link: http://www.math.toronto.edu/courses/apm346h1/20169/PDE-textbook/Chapter8/S8.P.html

I have a partial differential equation for my function $\Phi(\phi)$ which depends on l and m (l is a natural number, |m| < l ). For the case when l=0, I have  $\Phi(\phi)=c_0$, which solves my PDE for any c0. Do we have
$\Phi(\phi)=c_1 cos(\phi) +c_2$ for l=1
$\Phi(\phi)=c_3 cos^2(\phi) +c_4 cos(\phi) +c_5 $ for l=2
$\Phi(\phi)=c_6 cos^3(\phi) +c_7 cos^2(\phi) +c_8 cos(\phi) +c_9 $ for l=3? What do I do now? Do I plug in each  $\Phi(\phi)$ into its corresponding PDE and find the constants? Thanks

[Victor Ivrii]

For some $l,m$ you need multiply by $\sin(\phi)$