Analytic solution of the equation $c\int_0^t x^{a-1}e^{-x}dx + (c+e^t)e^{-t}t^{a-1} = 0$

Question

I would like to find the closed form solution of the equation in the title for the parameter $t$ when $-1<c<0$ and $0<a<1$.

I tried to use the Laplace transform. The transformation of both sides leads to the following:

$ s^{-a}+c(1+s)^{1-a}s^{-1} = 0$

which can be solved for $s$ :

$s=(1-(-c)^{\frac{1}{1-a}})^{-1}-1$

Is there a way to find the solution of the original equation using this or any other method to solve the problem?

Many thanks!

Answer 1

I think I got it.

hint: if you derivate the left part, the derivative is alot simpler. Then you can probably reintegrate, and fit the integration constant, thus getting a simpler equivalent equation to solve.