## Question on non homogeneous heat equation.

$\displaystyle \frac{\partial{u}}{\partial{t}}=\frac{\partial^2{u }}{\partial{r}^2}+ \frac {2}{r} \frac {\partial{u}}{\partial{r}}+\frac{1}{r^2}\left[\frac{\partial^2{u}}{\partial{\theta^2}}+\cot\thet a \frac{\partial{u}}{\partial {\theta}} +\csc\theta\frac{\partial^2{u}}{\partial{\phi}^2} \right]+q(r,\theta,t)$

Where $\displaystyle 0<r<a,\;0<\theta<\pi,\;0<\phi<2\pi,\;t>0$

with Boundary condition $\displaystyle u(a,\theta,\phi,t)=0$ and initial condition $\displaystyle u(r,\theta,\phi,0)=f(r,\theta,\phi)$.

I understand how to get to

$\displaystyle u(r,\theta,\phi,t)=\sum_{j=1}^{\infty}\sum_{n=0}^{ \infty}\sum_{m=-n}^{n} B_{jnm}j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)e^{-\lambda^2_{n,j} t}$

What I don't understand is the next step, the book assume

$\displaystyle g(r,\theta,\phi)=\sum_{j=1}^{\infty}\sum_{n=0}^{\i nfty}\sum_{m=-n}^{n} g_{jnm}j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)e^{-\lambda^2_{n,j} t}$

Where $\displaystyle g_{jnm}$ is another constant.

How do you justify to assume $\displaystyle g(r,\theta,\phi)$?