Farey sequence

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:

$\mathcal{ℒ}\{f^{\prime }\}=s\mathcal{ℒ}\{f\}-f(0)$

$\mathcal{ℒ}\{f^{″}\}=s^2\mathcal{ℒ}\{f\}-sf(0)-f^{\prime }(0)$

One by induction can prove that

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

Now we consider the following differential equation:

$\sum _{i=0}^n{a}_i{f}^{(i)}(t)=\varphi (t)$

with given initial conditions

$f^{(i)}(0)=c_i$

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

$\sum _{i=0}^n{a}_i\mathcal{ℒ}\{f^{(i)}(t)\}=\mathcal{ℒ}\{\varphi (t)\}$

obtaining

$\mathcal{ℒ}\{f(t)\}\sum _{i=0}^n{a}_i{s}^i-\sum _{i=1}^n\sum _{j=1}^i{a}_i{s}^{i-j}{f}^{(j-1)}(0)=\mathcal{ℒ}\{\varphi (t)\}$

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

$\mathcal{ℒ}\{f(t)\}=\frac{\mathcal{ℒ}\{\varphi (t)\}+\sum _{i=1}^n\sum _{j=1}^i{a}_i{s}^{i-j}{c}_{j-1}}{\sum _{i=0}^n{a}_i{s}^i}$

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.

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

then the formula simplifies to

$f(t)={\mathcal{ℒ}}^{-1}\left\{\frac{\mathcal{ℒ}\{\varphi (t)\}}{\sum _{i=0}^n{a}_i{s}^i}\right\}$

An example

We want to solve

$f^{″}(t)+4f(t)=\sin (2t)$

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

We note that

$\varphi (t)=\sin (2t)$

and we get

$\mathcal{ℒ}\{\varphi (t)\}=\frac{2}{s^2+4}$

The equation is then equivalent to

$s^2\mathcal{ℒ}\{f(t)\}-sf(0)-f^{\prime }(0)+4\mathcal{ℒ}\{f(t)\}=\mathcal{ℒ}\{\varphi (t)\}$

We deduce

$\mathcal{ℒ}\{f(t)\}=\frac{2}{(s^2+4{)}^2}$

Now we apply the Laplace inverse transform to get

$f(t)=\frac{1}{8}\sin (2t)-\frac{t}{4}\cos (2t)$

Bibliography

• 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