In this section we will be searching how to utilize Laplace transforms to solve differential equations. There are various types of transforms out there into the world. Laplace transforms and Fourier transforms are probably the major two types of transforms which are used. When we will see in shortly sections we can use Laplace transforms to decrease a differential equation to an algebra problem. The algebra can be messy on time, but this will be easy than in fact solving the differential equation directly in various cases. Laplace transforms can also be used to resolve IVP's which we can't use any previous method on.
For "simple" differential equations as those in the first only some sections of the last section Laplace transforms will be messier than we require. Actually, for most homogeneous differential equations as those in the last section Laplace transforms is considerably longer and not so helpful. Also, many of the "simple" non-homogeneous differential equations which we saw in the Undetermined Coefficients and Variation of Parameters are even simpler or at the least no more complicated than Laplace transforms to do as we did them there. Though, at this point, the amount of work needed for Laplace transforms is starting to equivalent the amount of work we did in those sections.
Laplace transforms arrives in its own while the forcing function in the differential equation starts finding more complicated. In the earlier section we searching for only at non-homogeneous differential equations wherein g(t) was a quite simple continuous function. Under this section we will start looking at g(t)'s which are not continuous. This is these problems where the cause for using Laplace transforms start to turns into clear.
We will also search that, for some of the more complex non-homogeneous differential equations from the last section, Laplace transforms are in fact easier on those problems also.