Solve the system of differential equations with by using Laplace transforms.

Hello!

For this problem, we need some properties of Laplace transform. They are:

`f'(t) -gt sF(s)-f(0)`

(here `F(s)` is the Laplace transform of `f(t)` ).

`sin(t) -gt 1/(s^2+1),`  `cos(t) -gt s/(s^2+1),`

`sinh(t) -gt 1/(s^2-1),`  `cosh(t) -gt s/(s^2-1).`

From these properties we obtain

`sX(s)=Y(s)+1/(s^2+1),`  `sY(s)=X(s)+(2s)/(s^2+1).`

It is a linear system for `X` and `Y` (the Laplace transforms of `x` and `y` ).

Its solution is

`X(s)=(3s)/((s^2+1)(s^2-1)) = 3/2 (s/(s^2-1)-s/(s^2+1)),`

`Y(s)=(2s^2+1)/((s^2+1)(s^2-1)) = 1/2 (3/(s^2-1)+s/(s^2+1)).`

Inverting Laplace transform we see that

`x(t)=3/2 (cosh(t)-cos(t)),`  `y(t) = 1/2(3sinh(t)+sin(t)).`

This is the answer.