Differential equations/Laplace transforms

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DefinitionEdit

For some problems, the Laplace transform can convert the problem into a more solvable form. The Laplace transform equation is defined as  , for the values of   such that the integral exists. There are many properties of the Laplace transform that make it desirable to work with, such as linearity, or in other words,   .

SolutionEdit

To illustrate how to solve a differential equation using the Laplace transform, let's take the following equation:   . The Laplace transform usually is suited for equations with initial conditions.

  1. Take the Laplace transform of both sides (  ).
  2. Use the associative property to split the left side into terms (  ).
  3. Use the theorem   , and by extension,   to modify the terms into scalars and multiples of   (  ).

  4. Solve for the Laplacian (  ).
  5. Take the inverse Laplace transform of both sides to get the solution, solving by method of partial fractions as needed:
    (  ).
  6. For reference, here are some basic Laplace transforms:
    1.  
    2.  
    3.  
    4.  
    5.  
    6.  
    7.  
  7. For reference, here are some theorems for the Laplace transforms:
    1.  
    2.  
    3.  
    4.