For $x>0$ let $\ f(x) = \int_0^\infty e^{-t-x^2⁄t} t^{-1/2}dt $
the question wants us to show that $\ f(x) = x \int_0^\infty e^{-t-x^2⁄t} t^{-3/2}dt $ by using substitution. However I do not think any substitution works here. What I have done so far is:
- found $f'(x)$ as $\ f'(x) = -2x ∫_0^\infty e^{-t-x^2⁄t} t^{-3/2}dt $ and
- using integration by parts i have, $\ f(x) = -2x^2 \int_0^\infty e^{-t-x^2⁄t} t^{-3/2}dt $
so that i get $\ f(x)= xf'(x) $ when i solve the differential equation i got $ f(x)= xe^C$ for some positive constant $C$ however the answer should be $ f(x)= Ce^{-2x} $ . I am totally wrong. Please help me. I can not go thorough..