Laplace transform applied to differential equations

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The Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions.

First consider the following property of the Laplace transform:


One can prove by induction that


Now we consider the following differential equation:


with given initial conditions


Using the linearity of the Laplace transform it is equivalent to rewrite the equation as




Solving the equation for  \mathcal{L}\{f(t)\} and substituting f^{(i)}(0) with c_i one obtains


The solution for f(t) is obtained by applying the inverse Laplace transform to \mathcal{L}\{f(t)\}.

Note that if the initial conditions are all zero, i.e.

f^{(i)}(0)=c_i=0\quad\forall i\in\{0,1,2,...\ n\}

then the formula simplifies to


An example[edit]

We want to solve


with initial conditions f(0) = 0 and f′(0)=0.

We note that


and we get


The equation is then equivalent to


We deduce


Now we apply the Laplace inverse transform to get



  • A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9