Question about separation of variables

Question

This is for the heat equation, where

$$\frac{\partial U}{\partial t}-k \frac{\partial^2 U}{\partial x^2}=1$$ with the conditions $$U(0,t)=0, \; U(x,0)=0 \text{ and } \frac{\partial U}{\partial t} (3,t)=0.$$

I am trying to solve for $U(x,t)$ but am currently stuck with factoring dealing with the "$+1$" in the separation of variables.

I started with $U(x,t)=F(x) G(t)$ then put it into the heat equation and set it equal to a constant -$\lambda^2$. To deal with the $+1$, I moved it to the other side with the lambda but now I am can't seem to get the sine or exponential expression I need.

Answer 1

Let $V=U-t$. Then

$$V_t-kV_{xx}=0.$$

Now do separation of variables (work out the appropriate boundary conditions for $V$ as well).

Answer 2

First assume the solution has the form

$$ u ( x,t ) = F_1 ( x ) F_2( t)+c_1+c_2x-\frac{1}{2k}\,x^2, $$

then advance with separation of variables technique.