## Question on non homogeneous heat equation.

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

Where $00$

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

I understand how to get to

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

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

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

Where $g_{jnm}$ is another constant.

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