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Solving Differential Equations Using the Laplace Transform

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 equationwith

Applying the Laplace transformation gives

Example: Solve the differential equationwith

Applying the Laplace transformation gives

Example: Solve the differential equationwithat

The transformed equation is

Solve the differential equationwithand

The transformed equation is

Hence

Applying the inverse transformation to each term gives