The Laplace transformation can be used to solve differential equations. Applying the Laplace transformation transforms differential equations into algebraic equations. We can solve the algebraic equation and apply the inverse transform from a table of Laplace transforms to obtain the solution to the original differential equation.
The transformation requires the formulae
and
Example: Solve the differential equation
with
Applying the Laplace transformation gives
Example: Solve the differential equation
with
Applying the Laplace transformation gives
Example: Solve the differential equation
with
at
The transformed equation is
Solve the differential equation
with
and
The transformed equation is
Hence
Applying the inverse transformation to each term gives