Laplace transform applied to differential equations explained

In mathematics, 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:

l{L}\{f'\}=s l{L}\{f\}-f(0) l{L}\{f''\}=s ^{2l{L}\{f\}-sf(0)-f'(0)}

One can prove by induction that

l{L}\{f ^{(n)} \} = s ^{n} _{i =1} s ^{n-i} f ^{(i-1)} ( 0 )

Now we consider the following differential equation:

\sum ^{n} _{i =0} a ^{(i)} _{if} (t) = \phi(t)

with given initial conditions

f ^{(i)} ( 0 ) = c _{i}

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

\sum ^{n} _{i =0} a ^{(i)} _{i l{L}\{f} (t)\} = l{L}\{\phi(t)\}

obtaining

l{L}\{f(t)\}\sum ^{n} _{i =0} a ^{n} _{i =1} \sum ^{i} _{j =1} a ^{i - j} _{is} f ^{(j-1)} ( 0 ) = l{L}\{\phi(t)\}

Solving the equation for

l{L}\{f(t)\} and substituting f ^{(i)} ( 0 ) with c _{i} one obtainsl{L}\{f(t)\}= (l{L)/(\{\phi(t)\}+\sum) ^{n} _{i =1} \sum ^{i} _{j =1} a ^{i - j} _{is} c _{j-1} }

The solution for f(t) is obtained by applying the inverse Laplace transform to

l{L}\{f(t)\}.

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

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

then the formula simplifies to

f(t) = l{L} ^{-1} \left\{{ l{L}\{\phi(t)\}\over\sum ^{n} _{i =0} a ^{i}\right\}} _{is}

An example

We want to solve

f''(t) +4 f(t) = \sin( 2 t)

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

We note that

\phi(t) = \sin( 2 t)

and we get

l{L}\{\phi(t)\}= (2)/(s ^{2+4})

The equation is then equivalent to

s ^{2l{L}\{f(t)\}-sf(0)-f'(0)+4l{L}\{f(t)\}=l{L}\{\phi(t)\}}

We deduce

l{L}\{f(t)\}= (2)/((s ^{2+4)} ^{2})

Now we apply the Laplace inverse transform to get

f(t) = (1)/(\sin( 2 t) - 8) (t)/(4) \cos( 2 t)

Bibliography

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