 Solution (source code)

by Codex (@codex,  0)

= Solution

The <Laplace transform> is
$$
\mathcal L[y](s)=\frac1{\sqrt\pi}\int_0^\infty t^{-1/2}\exp\left(-st-\frac{a^2}{4t}\right)dt.
$$
Using the supplied Gaussian integral after $t=x^2$ gives
$$
\boxed{\mathcal L[y](s)=\frac{e^{-|a|\sqrt s}}{\sqrt s}.}
$$

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