Suppose one is trying to solve this equation,
$$ \frac{\partial^2 U}{\partial x^{2}} + \frac{q}{\kappa} = \frac{\partial U}{\partial t} $$
why is it that we seek solutions of the form
$$U(x,t) = \Psi (x,t) + \phi(x)$$
I am trying to understand the underlying motivation for seeking solutions of this form, also, why is it that one cannot directly apply separation of variables to inhomogeneous PDE's. Thanks for your replies in advance.
Edit: if any of you know the physical difference between $\Psi(x,t)$ and $\phi(x)$, I would very much appreciate it if you could explain it.
$$ \frac{\partial^2 U}{\partial x^{2}} + \frac{q}{\kappa} = \frac{\partial U}{\partial t} \tag 1 $$ Try separation of variables : $\quad U(x,y)=F(x)G(y)$
$\begin{cases}\frac{\partial^2 U}{\partial x^{2}}=F''(x)G(y)\\ \frac{\partial U}{\partial t}=F(x)G'(t)\end{cases} \quad\to\quad F''(x)G(y)+\frac{q}{\kappa}=F(x)G'(t)$ $$\frac{G'(t)}{G(t)}-\frac{F''(x)}{F(x)}=\frac{q}{\kappa F(x)G(t)}$$
There is a function of $x$ and $t$ in the right hand term which cannot be separated into a sum of two functions, one of $x$, the other of $y$. This makes impossible the separation. The method fails.
So, we seek solutions of the form $$U(x,t) = \Psi (x,t) + \phi(x)$$ which transforms the PDE (1) into $$ \frac{\partial^2 \Psi}{\partial x^{2}} +\phi''(x)+ \frac{q}{\kappa} = \frac{\partial \Psi}{\partial t}$$ Let $\quad \phi''(x)=- \frac{q}{\kappa} \quad\to\quad \phi(x)=- \frac{q}{2\kappa}x^2 +bx+c$ $$ \frac{\partial^2 \Psi}{\partial x^{2}} = \frac{\partial \Psi}{\partial t}$$ Try again the separation of variables : $\quad \Psi(x,y)=F(x)G(y)$
$\begin{cases}\frac{\partial^2 \Psi}{\partial x^{2}}=F''(x)G(y)\\ \frac{\partial \Psi}{\partial t}=F(x)G'(t)\end{cases} \quad\to\quad F''(x)G(y)=F(x)G'(t)$ $$\frac{G'(t)}{G(t)}-\frac{F''(x)}{F(x)}=0$$ Now, it is possible to separate $$ \frac{G'(t)}{G(t)}=\frac{F''(x)}{F(x)}=\text{constant}$$ and to solve separately the two ODEs.