$K(X,T)=\frac{1}{\sqrt {4\pi T}}e^\frac{-x^2}{4T}$ is a solution for the equation $K_T = K_{XX}$ show that for any given function $g\in C^{\infty}$

Question

some kind of cauchy heat problem, need a hint please.

$K(X,T)=\frac{1}{\sqrt {4\pi T}}e^\frac{-x^2}{4T}$

is a solution for the equation $K_T = K_{XX} $.

show that for any given function $g\in C^{\infty}$.

$u(x,t) = e^{\frac{t^3}{3} -xt} \int _{\infty} ^{\infty} \ K(x-t^2-y, t) g(y)dy$

is a solution for the equation: $u_t +xu = u_{xx}$

I tried placing $K(X,T)$ in $u(x,t)$ and than placing $u(x,t)$ in the last equation but its kind of a mess and I could not get anything clear out of it. is there another way? thanks