2. This problem asks you to examine two numerical methods for solving differential equations. The techniques are Euler's and Improved Euler's methods.

a. Consider the differential equation

First, find the solution to this differential equation using either Maple (dsolve) or techniques that we develop in class. Are there any limits on the domain of this solution?

b. Apply the Euler's and Improved Euler's methods to this problem with a stepsize of h = 0.1 for t [0,1], then repeat the process with h = 0.05. Create a table with the values of the actual solution, the approximate solutions, and the percent errors (when compared to the actual solution) at t = 0.5 and 0.9 for both stepsizes and both methods.

c. Sketch the graphs of the true solution and each of the approximate solutions with h = 0.1. Be sure to label which graph corresponds to which method.

d. Now consider the differential equation:

where a = 1.2 and b = 0.5. Find the solution to this differential equation using techniques from class or Maple's dsolve. Find when the solution has a maximum.

e. Apply the Euler's and Improved Euler's methods to this problem with a stepsize of h = 0.5 for t [0,6]. Create a table with the values of the actual solution, the approximate solutions, and the percent error (when compared to the actual solution) at t = 1, 3, and 5.

f. Sketch the graphs of the true solution and each of the approximate solutions from Part e. Be sure to label which graph corresponds to which method.