Exercises

Exercise 1. The initial value problem , was used as an initial example of Euler's method. In this case the exact solution is known (what is it?), so we are in a position to compare the approximate solution generated by Euler's method with the exact value. Repeat the steps carried out in the section on Euler's Method in generating tables and plots for this problem on the interval [0, 1]. The exact value of is 5. Explain why all the approximate solution curves fall short of this value. (Hint: note the concavity of the function .) Generate solutions of the problem corresponding to smaller and smaller values of , for example = 0.1, 0.01, 0.001. Do the values of approach the exact answer 5 as gets smaller?

Exercise 2. Find the exact solution to the initial-value problem , . Then apply Euler's Method to plot the solution on the interval [0, 10]. Where, approximately, is the first positive zero of the solution? Can you find the solution more exactly? Hint: Use Maple's fsolve function, applied to the exact solution you found for this problem.