Derive the following formula
$u(t,x)={1 \over 3}t^3+{1 \over 2}t^2x^2$
for the one-dimensional non-homogeneous heat equation
$u_t=u_{xx}+tx^2, t>0, x\in (-\infty,\infty)$
$u(0,x)=0.$
(without plug in)
I tried Fourier Transform but nothing helped since
$\int e^{-iwx} tx^2 dx$ is not integrable.
Please help.
Use laplace transform on t $U'=\frac{\partial U(s,x)}{\partial x}$
$$U''(x)-s U(x)=-\frac{x^2}{s^2}$$
$$U(s,x)=c_1 e^{\sqrt{s} x}+c_2 e^{-\sqrt{s} x}+\frac{s x^2+2}{s^4}$$
Applying initial conditions (https://en.wikipedia.org/wiki/Initial_value_theorem) using $s\rightarrow \infty$ and keeping in mind that we have both positive and negative $x$
$$U(s,x)=\frac{s x^2+2}{s^4}$$
Inverse laplace transform gives us
$$\frac{t^3}{3}+\frac{t^2 x^2}{2}$$