Example 10. Use Mathematica to find the analytic solution and graph for the I.V.P.  .  

Solution 10.

Dig out the formula for the solution out of the data structure of  solset and put it in  f[t].
Plot the analytic solution at the same sample points that were used for the numerical approximations.

Just for fun, plot the Heun solution and the analytic solution. Notice that there is a difference.

Something strange is happening, it appears that the solution to  has a vertical asymptote, yet Heun's method is able to move past this asymptote!  How can this happen?  If the step size is chosen smaller, then these methods will not tend to overshoot the asymptote. The location of the asymptote can be found using Mathematica's procedure FindRoot.

Numerical methods with a higher order of precision will not tend to overshoot the asymptote.

(c) John H. Mathews 2004