Consider the equation $u_t = u_{xx} - 3 u$, $0 < x < 1$, $t > 0$, with boundary condition $u(0,t) = 0 , u(1,t) = 6 \sinh(\sqrt{3})$, and initial condition $u(x,0) = 6\sinh\!\left(\sqrt{3}x\right)+8\sin\!\left(2\pi x\right)$.

We wish to solve it. Do this in two steps: first make the boundary conditions homogeneous, then get rid of the heat loss term. It may be useful to start by solving for the steady state.

$u_{\text{steady state}}(x) = {}$

Then the final solution is
$u(x,t) = {}$

help (formulas)