How to find Laplace transform of $\frac{1}{\sqrt{\pi t}} e^{at} (1+2at) $

Question

My textbook says the answer is $\frac{s}{(s-a)^{\frac{3}{2}} }$ but I am having difficulties getting even close to an expression like that. I tried to distribute the product and then first evaluate the Laplace transform of $$ \frac{1}{\sqrt{\pi t}} e^{at}$$ by making the substitution $t=u^{2}$. By doing that I got $$\int_{0}^{\infty} \frac{1}{\sqrt{\pi t}} e^{at} e^{-st}= e^{s-a} $$ Now, I am not sure about the other piece and I fear this might not be the right approach to get the answer shown in the textbook. Can you please show me how to proceed?

Answer 1

Your first evaluation is wrong, maybe this hint will help: Note that $$ \int_0^\infty f(t) e^{a t} e^{-s t} dt = \int_0^\infty f(t) e^{-(s - a)t} = \int_0^\infty f(t) e^{-b t} \ dt = \mathcal L[f(s-a)]. $$ Also, $$ \int_0^\infty t^\alpha e^{-st} \ dt = \frac{\Gamma(1+\alpha)}{s^{1+\alpha}}. $$ Setting $\alpha = -1/2$ and combining yields: $$ \frac{1}{\sqrt{s-a}}. $$