# Applying the Laplace transform to solve Differential equations

• A
• LagrangeEuler
In summary, a Laplace transform can be applied to a differential equation of finite order to obtain an algebraic equation. This is only valid for differential equations with constant coefficients. An example of this transformation is the equation ##y''(t)+\sin(t)y(t)=0## with initial conditions ##y(0)=A## and ##y'(0)=B##. However, it is not clear why one would want to do this or if it is valid in all cases.

#### LagrangeEuler

Is it possible to apply Laplace transform to some equation of finite order, second for instance, and get the differential equation of infinite order?

A laplace transform turns a differential equation into an algebraic equation.

If you had a term like ##\sin t\,y##, you could expand ##\sin t## as a series and take the Laplace transform of the result term by term, which would give you a bunch of derivatives of Y(s). I'm not sure why you'd want to do that though or if doing so is valid.

pasmith said:
A laplace transform turns a differential equation into an algebraic equation.
It is only in the case when you have a differential equation with constant coefficients.

Do you have an example in mind?

##y''(t)+\sin(t)y(t)=0##, where ##y(0)=A##, ##y'(0)=B##. If I apply the Laplace transform would I get the differential equation of infinite order?