Background. Numerical methods are useful in constructing solutions to differential equations.
For illustration, we consider Euler's method.

Algorithm. Euler's Method. To approximate the solution of the initial value problem
with over by computing
for .

Computer Lab Work.

Exercise 1. Consider the initial value problem with y(0) = 1 over [0, 0.95].
Use Euler's method and compute and graph a numerical solution.
(a) Use 19 steps of size h = 0.05 , include the list of points.
(b) Use 38 steps of size h = 0.025 , include the list of points.
(c) Use 76 steps of size h = 0.0125 , omit the list of points.
(d) Use 152 steps of size h = 0.00625 , omit the list of points.
(e) Observe that the sequence of solutions (a)-(d) appear to be converging.

Exercise 2. Consider the initial value problem with y(0) = 1 over [0, 0.95].
Use Mathematica's built in procedure NDSolve to compute a numerical solution, and then
use Mathematica's Evaluate function to plot the solution
Observe that we need not be aware how the computations are done.

Exercise 3. Consider the initial value problem with y(0) = 1 over [0, 0.95].
Continue this investigation only if you feel comfortable with someone telling you that the
"analytic solution" is represented with the following "special function."

Exercise 4. For those who are curious, we can do a little investigation regarding this solution.
First, an exploration regarding the Gamma function (which is o.k. for x > 0).

Second, check out the initial value for the proposed solution.

Third, check out the location of the vertical asymptote of the proposed solution.

Fourth, like Tan[t], the function f[t] is actually defined past the asymptote.

You don't need to worry about the fractional Bessel function in this course.

Solutions.

Return to the Differential Equations Project

Return to the Numerical Analysis Project

Return to the Complex Analysis Project

(c) John H. Mathews, 1998