The separation of variables in a non-homogenous equation (theory clarification) I know "copying and paste" method from resources aren't permitted but the text is fairly long and given the amount of time I have to learn PDE (as an exchange student beside having to adapt to a different learning style) is fairly short, with a steep learning curve, my justification is that I do not expect myself to labour through a wall of LaTeX.
Time is essentially a luxury to me.

Observe that the given PDE heat equation is:
$$\frac{\text{$\delta $u}}{\text{$\delta $t}}\text{=3}\frac{\delta ^2u}{\text{$\delta
   $x}^2}\text{+x}$$
We know that in order to solve for $$V(x,t)$$, 
We set $$V(x,t)$$ to the form equivalent to the given non-homogenous heat PDE, that is,
$$\frac{\text{$\delta $v}}{\text{$\delta $t}}\text{=3}\frac{\delta ^2v}{\text{$\delta
   $x}^2}$$ (without the x) and then we perform the method of separation of variables.
My question is why do we do this? The above equation is essential a juxtapose of the heat equation with "U" replaced with "V" (without the x). Is the reason grounded in physical intuition and/or mathematica convenience? I would appreciate simple explanation (if possible).
 A: The answer to your question is so-called superposition principle of linear differential equations, which says that if $u_1(x)$ and $u_2(x)$ are two distinct solutions of a linear PDE, then their linear combination $u_3 (x)= a\,u_1(x)+b\,u_2(x)$ will also be a solution of the PDE.
This principle result in the following technique of solving inhomogeneous PDEs 

Assume we have an inhomogeneous  Partial Differential Equation of the form 
  $$  Au_{xx} + 2Bu_{xy}+Cu_{yy} + Du_{x} + Eu_{y} + Fu = w(x, y)\tag{1}\label{1}$$
  with some initial and boundary conditions. 
Let us define theauxiliary linear homogeneous equation as
  $$  Av_{xx} + 2Bv_{xy}+Cv_{yy} + Dv_{x} + Ev_{y} + Fv = 0 \tag{2}\label{2} $$
  with the same boundary conditions as in $\eqref{1}$
Then the general solution $u = u^{\text{general}}$ of the inhomogeneous equation $\eqref{1}$ can be written as the sum of a particular solution $u^{\text{particular}}$ of $\eqref{1}$ and a general solution $v^{\text{general}}$ of the auxiliary homogeneous equation$\eqref{2}$:
  $$ u^{\text{general}} (x,y)  = v^{\text{general}} (x,y)  +  u^{\text{particular}} (x,y) .$$

This principle holds for linear partial and ordinary differential equations of any order.
Here  are several examples of solving inhomogeneous heat equation using superposition principle.
