# Partial Differential Equations: Finding Explicit expression

Find an explicit expression for the solution to

$\partial_{t}u=\partial_{xx}u+10u\\ u(x,0)=cos(x)+cos(3x)$

So I know how I would do this is if it was $10$ instead of $10u$ in the first equation. So if anyone could just help me with how to account for the $10u$ that would be lovely.

if we make a change of variable $$u = ve^{10t}, u_t = (v_t+10v)e^{10t}, u_{xx} = v_{xx}e^{10t}$$ with this we have $$v_t = v_{xx}, v(x,0) = \cos x + \cos 3x.$$ the solution is $$v =\cos x e^{-t}+ \cos 3x e^{-9t},\, v =e^{9t}\cos x+ e^{t}\cos 3x,$$