In this problem we investigate Picards method for approximating the solutions of

Chapter 0, Problem 59

(choose chapter or problem)

In this problem we investigate Picards method for approximating the solutions of differential equations. Consider the differential equation y (t) = y(t) 2 + t 2 , y(a) = b. Integrating both sides with respect to t gives y(s) y(a) = , s a y (t) dt = , s a (y(t) 2 + t 2 ) dt. Since y(a) = b, we have y(s) = b + , s a (y(t) 2 + t 2 ) dt. We have put the differential equation into the form of an integral equation. If we have an approximate solution y0(s), we can use the integral form to make a new approximation y1(s) = b + , s a (y0(t) 2 + t 2 ) dt. Continuing this process, we get a sequence of approximations y0(s), y1(s), ..., yn(s), . . . where each term in the sequence is defined in terms of the previous one by the equation yn+1(s) = b + , s a (yn(t) 2 + t 2 ) dt. (a) Show that yn satisfies the initial condition yn(a) = b for all n. (b) Using the initial condition y(1) = 0, start with the approximation y0(s)=0 and use a computer algebra system to find the next three approximations y1, y2, and y3. (c) Use a computer algebra system to find the solution y satisfying y(1) = 0, and sketch y, y1, y2, y3 on the same axes. On what domain do the approximations appear to be accurate? [The solution y cannot be expressed in terms of elementary functions. If your computer algebra system cannot solve the equation exactly, use a numerical method such as Eulers method.]