How come the following equality for the integral

Question

There is a function $h(t)=\int\limits_0^{+\infty}\exp\left\{- \frac{t^2}{x^2}-x^2 \right\}dx, t>0$

It's stated that $h'(t)=-2h(t)$.

But $h'(t)=-2\int\limits_0^{+\infty}\exp\left\{- \frac{t^2}{x^2}-x^2 \right\}\frac{t}{x^2}dx$ and i don't know how to get the result above.

Could you give me a hint please?

Answer 1

Make the substitution $u = \frac{t}{x}$.