Recall that the Laplace Transform is a mapping that ‘eats’ functions of single (positive real) variable and ‘spits out’ functions of a single (complex) variable . If we write for the set of functions of a single positive real variable and for the set of functions of a single complex variable then we might write

.

In this example we could have been more careful and explicit and wrote that is the solution of the differential equation.

Now finally, rather than carry around the messy — the Laplace Transform of (in this case the Laplace Transform of the solution of the differential equation), we just use the notation

.

Regards,

J.P.