How Do I Solve a Heat Equation with Unknown Forcing Term p(x, t)?

[Jamin2112]

Homework Statement

v_t(x,t)=v_xx(x,t) + p(x,t),
Neumann boundary conditions,
v(x,0)=cos(∏x)

Homework Equations

Assume v(x,t)=X(x)T(t)

The Attempt at a Solution

I'm stuck. We aren't given a p(x,t) and I'm not sure what to do. Where do I go from here?

Attempt so far:

 

[Jamin2112]

So I got a little farther ...

The full solution is

e^-n2∏2tcos(∏x) + Ʃe^-n2∏2t(∫_[0,t]p_n(t)e^n2∏2tdt)cos(n∏x).

But I still can't figure out any more information without knowing what exactly p(x,t) is. (Right?) I'm asked to solve the equation and then explain "For what forcing does the temperature eventually settle down to a constant." Thoughts?


EDIT: Also, I know that p_n(t)=∫_[0,1]p(y,t)cos(n∏y)dy, though I can't figure this out (Can I?) unless I'm explicitly told what p(y,t) is.