Let $f (x)=x^2e^{-x} $. Determine the solution $u (x,t) $ of the system of equations$$\partial^2_xu=\frac 1 k \partial_tu$$ $$u (x,0)=f (x),$$ for a constant $k $.
Is there a particular way to resolve this problem? I don't have anything similar in my notes, and I didn't have an intuition by myself. Thanks for any explanation.
Hint: $$ \begin{cases} \partial^2_xu=\frac 1 k \partial_tu \\ u (x,0)=f (x) \end{cases} $$ For the Heat's Equation, you can use what you have recently learned, the Fourier Transform. $$ \begin{cases} \hat u''(\omega,t)-i\omega \frac 1 k\hat u(\omega,t) =0 \\ \hat u (\omega,0)=\hat f (\omega) \text {, where } \hat f (\omega)=\omega ^2 e^{-\omega} \end{cases} $$ Now you have a second order linear differential equation. you should be able to solve it.