Solve the Integral Equation Involving Laplace Transforms

Question

I want to solve $\int^\infty_0x'(T)x(t-T)dT=6t^3$ where $x(0)=0$

I did the Laplace transform to both sides, and the left side is a convolution, so I then have

$X(s)x(s)=\frac{36}{s^4}$, but here I'm stuck. Any help to move forward? Thanks!

Answer 1

The LHS of the equation is, after LT, $s X(s)^2$. Thus,

$$X(s) = 6 s^{-5/2}$$

According to this table, the ILT is

$$x(t) = \frac{4}{\sqrt{\pi}} t^{3/2} $$

If I have time later, I will derive this result via contour integration in the complex plane.