The Laplace Transform We have seen many techniques of solving differential equations that involve using a substitution. There is a special type of substitution, called an integral transform that simplifies the task of solving differential equations.
Note: a and b can be any real numbers or even infinity or negative infinity The most important integral transform in the field of differential equations is when a is 0, b is infinity, and K(s,t) is e^{st}.
Example Find the Laplace Transform of f(x) = x^{2}.
Solution We just work out the integral
This can be worked out by integrating by parts twice.
We used the fact that
for all s > 0. This fact is easily proven using induction and LHopital's rule. We can see that finding the Laplace transform of a function us just a matter of integration. We will usually see integration by parts enter into the picture. Exercises Find the Laplace transform of the following functions A. f(x) = sin(at) B. f(x) = cos(at) C. f(x) = e^{at} D. f(x) = t^{n} n a positive integer E. f(x) = e^{at} sin(bt)
Our next question to ask is when the Laplace transform of a function is defined. Since the Laplace transform of a function is defined as an improper integral, the integral may not converge. Fortunately most of the functions that we know and love have convergent Laplace transforms. More generally we have the following theorem.
Proof First, since f(t) is piecewise continuous, for large values of t, f(t) is continuous. Since convergence only depends on the behavior for large values of t, we can assume that f is continuous. To finish the proof we substitute the larger Ke^{at} for the smaller f(t) to get
This converges whenever a < s.
Another important fact about Laplace Transforms is that the Laplace transform is a linear transformation from "nice" functions to functions, where "nice" means functions that have a Laplace transform. That is, L{f(t) + g(t)} = L{f(t)} + L{g(t)} and L{cf(t)} = cL{f(t)}
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