Today we were given this general solution (before solving for T with the inverse Fourier transformation) $$\hat T(\sigma,t) = \int_0^t e^{-\sigma^2(t-s)}\hat q(\sigma,s) ds$$ for the heat equation with internal heat source q formulated as $$ HP : \begin{cases} T_t = T_{xx} + q(x,t) & (x,t)\in D \\ T(x,0) = 0 & x \in \mathbb{R} \end{cases}$$ where $D = {(x,t)\in \mathbb{R}^2 : t > 0}$. We used Fourier transformation under the hypothesis of $\hat T$ and its derivate being confined by a function of $\sigma $ to obtain: $$ HP : \begin{cases} \hat T_t(\sigma,t) + \sigma^2 \hat T(\sigma, t) = \hat q(x,t) & \\ \hat T(\sigma,0)=0 & \end{cases}$$ ($\kappa = 1$ to simplifie)
UPDATE: a friend told me it has to do with the "Integration factor method". I tried to apply it but still do something wrong.
I'cant see how this solution was obtained. Can anybody show me? Thank you very much!
So it is a application of the integrating factor (https://en.wikipedia.org/wiki/Integrating_factor) with $$M(\sigma,t)=e^{\int_{s_0}^{t}\sigma^2 dy }$$ since in the ODE we have $\sigma^2$ as factor to $\hat T$ which doesn't depend from our dynamic variable t. We can than chose $s_0$ in and out as we want to fit the proposition above because we are just searching for one solution.