Checking the solution of a differential equation with a boundary condition depending on an arbitrary function

Question

I have the partial differential equation:

$\qquad U_{x} - 4 x^{2} U_{t} = x U, \qquad U(x,0)=f(x), \qquad x \in (- \infty, \infty), \qquad t \in (0, \infty)$

Where $U_{x} = \frac{\partial U}{\partial x}$, and the solution I arrived at using Method of Characteristics is:

$\qquad U(x,t) = f\Big( (x^{3}+\frac{3}{4}t)^{\frac{1}{3}} \Big) * e^{\frac{x^{2}}{2} - \frac{1}{2} (x^{3} + \frac{3}{4}t)^{\frac{2}{3}}}$

I am unsure how to define the arbitrary function $f(x,t)$ in that way when trying to plug the solution back into the differential equation to check to see if it is a solution. (This solution may not be correct)

Answer 1

In case reading the documentation, as recommended by yohbs, did not fully answer your question, try

U[x_, t] = f[(x^3 + 3 t/4)^(1/3)] Exp[x^2/2 - (x^3 + 3 t/4)^(2/3)/2];
Simplify[D[U[x, t], x] - 4 x^2 D[U[x, t], t] - x U[x, t]]
(* 0 *)

By the way, Exp[ - (x^3 + 3 t/4)^(2/3)/2] can be absorbed into f without loss of generality.