Laplace Transform of Sine of Root/Proof 2

Proof
Let $\map y t := \sin \sqrt t$.

Differentiating twice $t$, we get:
 * $(1): \quad 4 t y + 2 y + y = 0$

Let $\map Y s = \laptrans {\map t y}$ be the Laplace transform of $y$.

Then taking the Laplace transform of $(1)$:

For small $t$, we have:
 * $\sin \sqrt t \sim \sqrt t$

and:
 * $\laptrans {\sqrt t} = \dfrac {\sqrt \pi} {2 s^{3/2} }$

For large $t$, we have:
 * $Y \sim \dfrac c {2 s^{3/2} }$

Hence by comparison:
 * $c = \dfrac {\sqrt \pi} 2$

Hence:
 * $\laptrans {\sin \sqrt t} = \dfrac {\sqrt \pi} {2 s^{3/2} } \map \exp {-\dfrac 1 {4 s} }$