Finding the Laplace transform (using the Table of Laplace Transforms) of each differential equation yields:

First equation:

...(1)

Second equation:

...(2)

We now have 2 equations in 3 unknowns (X, Y and s). We need to solve for X and Y and leave s on the right hand side of each expression. We choose to solve for Y first.

(2) × (s + 1):

...(3)

(1) − (3) gives:

The last step above is using partial fractions.

s² terms: A + B = 3

s terms: 2A + C = 7

Constant terms: 5A = 10

So

We can now write Y as the sum of partial fractions:

Finding the inverse Laplace of our expression for Y gives:

We now need to find the expression for x. Equation (2) gives us:

and since , we have:

So

Therefore,