[0,1],
then repeat the process with h
= 0.01. 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. 1. This problem asks you to examine Euler's method for solving differential equations.
a. Consider the differential equation
First, find the symbolic solution to this differential equation using (a) Maple (dsolve) and (b) by-hand techniques that we developed in class. Are there any limits on the domain of this solution?
b. Apply Euler's method to this problem with a
stepsize of h = 0.1
for t
[0,1],
then repeat the process with h
= 0.01. 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.
c. Sketch the graphs of the true solution and the approximate solution with h = 0.1. Be sure to include the location of the vertical asymptotes and label which graph corresponds to the numerical solution.
2. Now consider the differential equation:
where a = 1.2 and b = 0.5.
a. Find the solution to this differential equation using techniques from class and Maple's dsolve. Show the steps in both derivations. Find when the solution has a maximum.
b. Apply Euler's method 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.
c. Sketch the graphs of the true solution and the approximate solution from Part b. Be sure to label which graph corresponds to which method.
3. Consider the differential equation:
Sketch the solution for t
[0,10]
and find y(10) correct to 4 significant figures by any method. Write
a couple sentences describing how you obtained your answer.