Answer to Question #127079 in Differential Equations for jse

Question

Using integration by parts, find the Laplace transform of the given function; a is a real constant.

f(t) = t^2 sinh(at)

Your answer should be an expression in terms of a and s.
Enclose numerators and denominators in parentheses. For example, (a−b)/(1+n).


L{f(t)}(s )= F(s )= ______________

Answer 1

$F(s)=\int\limits_0^\infty e^{-t\cdot s}\cdot f(t)\,dt=\\ =\int\limits_0^\infty e^{-t\cdot s}\cdot t^2\cdot\sinh(at)\,dt=\\ =\int\limits_0^\infty e^{-t\cdot s}\cdot t^2\cdot\frac{e^{at}-e^{-at}}{2}\,dt=\\ =\frac{1}{2}\cdot\int\limits_0^\infty t^2\cdot(e^{(-s+a)t}-e^{(-s-a)t})\,dt=\\ =\frac{1}{2}\cdot\int\limits_0^\infty t^2\cdot e^{-(s-a)t}\,dt-\frac{1}{2}\cdot\int\limits_0^\infty t^2\cdot e^{-(s+a)t}\,dt=\\ =\frac{1}{2}\cdot \frac{1}{(s-a)^3}\cdot\varGamma(3) - \frac{1}{2}\cdot \frac{1}{(s+a)^3}\cdot\varGamma(3)=\\ =\{\varGamma(3)=(3-1)!=2\}=\frac{1}{(s-a)^3}-\frac{1}{(s+a)^3};$