Ordinary Differential Equations/Laplace Transform

DefinitionEdit

Let   be a function on  . The Laplace transform of   is defined by the integral

 

The domain of   is all values of   such that the integral exists.

ExistenceEdit

PropertiesEdit

LinearityEdit

Let   and   be functions whose Laplace transforms exist for   and let   and   be constants. Then, for  ,

 

which can be proved using the properties of improper integrals.

Shifting in sEdit

If the Laplace transform   exists for  , then

 

for  .

Proof.

 

Laplace Transform of Higher-Order DerivativesEdit

If  , then  

Proof:
 
 
  (integrating by parts)
 
 
 

Using the above and the linearity of Laplace Transforms, it is easy to prove that  

Derivatives of the Laplace TransformEdit

If  , then  

Laplace Transform of Few Simple FunctionsEdit

  1.  
  2.  
  3.  
  4.  
  5.  
  6.  

Inverse Laplace TransformEdit

DefinitionEdit

LinearityEdit