Problems

1. Show that Euler's method fails to approximate the solution of the initial value problem

   ,  , . Explain.

2. Determine analytically the Euler approximation to the initial value problem

   , .

   Find also the exact solution of the problem and determine the magnified error function.
3. How large is the discretization error of the approximation to the solution of the initial value problem

    

   obtained by Euler's method?

4. What step size would you use with Euler's method to integrate from to in order to achieve errors (ignoring round off errors) of not more than the following:

   •  0.1

   •  0.01

   •  0.001

5. Consider solving the following initial value problem by Euler's method

   •  

   • 

   What step size would one use to achieve an error (ignore round off errors) less than

   0.01?

6. Determine error bounds (a priori, a posteriori, and an error estimate) when solving the

   following initial value problems over by Euler's method

   • 

   • 

   • ,

7. Determine the magnified error function for the numerical solution of the initial value

   problem

   ,

   by the Euler's method.