Solving Nonhomogeneous Heat Equation with Fourier Transform

[ColdFusion85]

How would one obtain a Fourier Transform solution of a non homogeneous heat equation? I've arrived at a form that has

$$\frac{\partial }{ \partial t }\hat u_c (\omega,t) + (\omega^2 + 1)\hat u_c (\omega,t) = -f(t)$$

My professor gave us the hint to use an integrating factor, but I don't see how this would work. If $$p(x) = (\omega^2 + 1)$$, and $$q(x) = -f(t)$$, then the integrating factor would be $$e^{\int(\omega^2 +1)dw}$$

Multiplying through by this would not give the usual integrating factor form of (..)' on the left hand side. Can anyone help?

 

[ColdFusion85]

Anybody?

 

[ramtin]

The integrating factor is :
$$e^{\int(\omega^2 +1)dt}$$

but I don't know what to do next