Laplace transform

Laplace transforms are one of the ways of solving linear ordinary differential equations (Linear ODEs) with constant coefficients. This technique allows us to transform a Linear ODE into a linear algebraic equation.

## Definition[edit]

The unilateral Laplace transform is defined by

Given the integral converges. A necessary condition for this integral to converge is

## Examples[edit]

Consider the following initial value problem

where  is constant, subject to

To solve this problem using laplace transform, first apply laplace transform to both sides of the equation, obtaining:

The integral on the right hand side is

If ,

For the left side, if we apply integration by parts,

Substitution into the left side will get

For the transformation to converge,

Therefore, substituting the initial condition y(0)=0 the left side becomes

Equating the two sides of the equation:

If , we can use partial fractions to changet the right side into

and the solution is obtained by noticing:

Substituting the inverse transform we have

Which leads to the answer

## References[edit]

* D. Lomen and D. Lovelock, Differential Equations Graphics. Model. Data., John Wiley and Sons, Toronto, 1999.
* Laplace transform on Wolfram Mathworld