Laplace transform of the sin hyperbolic function

Submitted by Structure on Sat, 12/08/2007 - 17:12.

Function $f:R\rightarrow R$ , $f(t)=\sinh t\:h(t)$ verifies differential equation

$x"(t)-x(t)=0\;\:x(0+)=0,\:\:x'(0+)=1$

$\mathcal{L}(x(t)(s)=X(s)$

$\mathcal{L}(x'(t)(s)=sX(s)$

$\mathcal{L}(x"\;(t)(s)=s^2X(s)-1$

We have

$s^2X(s)-1-X(s)=0$

$X(s)=\mathcal{L}(\sinh t\:h(t))(s)=\frac{1}{s^2-1}$