Laplace Transform for e^(at)

The equation above yields what the Laplace Transform is for any function of the form `e^(at),` where `a` is an arbitrary scalar.
     `cc"L" [e^(at)] = frac(1)(s-a) (s>a)`

In general, Laplace Transforms "operate on a function to yield another function" (Poking, Boggess, Arnold, 190).
Given a function `f(t)` from `0<t<infty`, the Laplace Transform is:
     `mathcal(L)(f)(s)=F(s)=int_0^infty f(t)e^(-st) dt " for " s>0`

To find the Laplace Transform of a function, `f(t)`, you have to evaluate an improper integral. We will not discuss how you would evaluate this type of integral, but rather we will discuss why Laplace Transforms are useful for differential equations. Since a Laplace Transform is taking a function and "transforming" it into another function, Laplace Transforms are valuable for finding solutions to differential equations that are made up of linear, continuous functions, or discontinuous functions. 

There are many uses for Laplace Transforms, but we are not going to go into heavy detail about all of them, rather we are going to state some basic rules regarding Laplace Transforms. 

The Laplace Transform of `frac(dy)(dt)` is as follows, when given a function y(t) with Laplace Transform `cc"L" [y]` :
     `cc"L"[frac(dy)(dt)]=s cc"L"[y]-y(0).` 

Laplace Transforms have linearity in the following way when applied to functions `f`, `g`, and constant `c`:
     `cc"L" [f+g]=cc"L"[f] +cc"L"[g]`
     `cc"L"[c*f]=c* cc"L"[f]`

If you would like to see how Laplace Transforms can be used to solve an initial-value problem, please visit our Laplace Transforms Calculator.

Laplace Transforms

• `mathcal{L}(t^{n})(s)`
• `\mathcal{L} (t^{n}e^{at}) (s)`
• `mathcal{L}(sin(at))(s)`
• `mathcal{L}(e^{at}sin(bt))(s)`
• `\mathcal{L} (e^{at}cos(bt)) (s)`
• `\mathcal{L} (cos(at)) (s)`
• `\mathcal{L} (e^{at}) (s)`

Sources

Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. Differential Equations. 3rd ed. Belmont, CA: Thomson Brooks/Cole, 2006. Print. 

Polking, John C., Albert Boggess, and David Arnold. "Mixing Problems." Differential Equations with Boundary Value Problems. Upper Saddle River, NJ: Pearson/Prentice Hall, 2006. N. pag. Print.