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.  

Definition of an Integral Transform

          Let K(s,t) be a function of two variables.  The the Integral Transform with Kernel K, is defined as the mapping that takes functions to functions by the rule


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.  

          Definition of the Laplace Transform

Let f(x) be a function.  Then the Laplace Transform of f(x) is 




Find the Laplace Transform of f(x)  =  x2



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.


Find the Laplace transform of the following functions

A.  f(x)  =  sin(at)

B.  f(x)  =  cos(at)

C.  f(x)  =  eat 

D.  f(x)  =  tn  n a positive integer

E.  f(x)  =  eat 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.


Let f(t) be a piecewise continuous function on 0 < t < A for any A Also let 

          |f(t)| < Keat       

for all t > M with K, a and M real constants and K and M positive.

Then the Laplace transform L{f(t)}  =  F(s) exists for s > a



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 Keat 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)}


        L{cf(t)}  =  cL{f(t)}


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