Solving the integral equation associated with Laplace transform

Question

I am clueless on how to solve this integral equation: $\lambda\int_{0}^{\infty}f(x)\exp\{-\lambda x\}dx=\sqrt{2\lambda}$, where function $f$ is a non-negative measurable function. And the result is $f(x)=\sqrt{\frac{2}{\pi x}}$. Any help would be appreciated. I originally want to solve it through differentiating both sides, but I am not sure how to do that.

Answer 1

Hint. Compare your integral equation with the definition of the Gamma function, $$ \Gamma(z)=\int_0^{\infty}t^{z-1}e^{-t}\,dt\qquad(\Re(z)>0), \tag{1} $$ from which follows $$ \int_0^{\infty}x^{z-1}e^{-\lambda x}\,dx=\frac{\Gamma(z)}{\lambda^z}. \tag{2} $$