Euler's Method for  Approximate Solution of First Order Differential Equations

 

 

In a Nut Shell:  Euler's Method provides a simple approach to obtain a numerical solution
of linear or nonlinear first order differential equations.  However, care must be exercised

since convergence is not guaranteed.  Numerical errors may accumulate leading to
erroneous results.

 

 

Problem Statement:    Find an approximate solution to the first order, ordinary d.e.

 

                    dy/dx   =  f [ (x,y(x) ]     subject to the condition        y(0)  =  yo

 

 

Strategy:    Use Euler's Algorithm to obtain successive approximations

 

                          yn+1  =  yn  +  h f(xn, yn)        ( n ≥ 0 )

 

where  h  =  step size

 

Of course smaller step sizes require more steps to arrive at a result whereas larger step

sizes may yield less accurate results.

 

 

Local and Cumulative Errors related to Euler's Method.

 

The linear approximation to the solution curve is as follows:

 

       y(xn+1)  ≈  yn  +  h f(xn, yn)  =  yn+1           The figure below shows the local error.

 

                                         

 

 

 

  Trade-off:  The local error can accumulate.  So selection of the step size, h, is an

                      Important consideration.

                                  

 

Click here for an example.

 

 

 



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