ODE_Series Solutions.html

Series Solutions

Differential Equations Project

Objectives:

(a) To understand the use of power series in approximating a solution to a differential equation

whose coefficients are functions of the independent variable.

(b) To introduce the series option of the dsolve command, as well as the commands Order

and convert .

The power series method will allow us to handle differential equations with variable

coefficients, most of which cannot be solved in closed form; but this method involves quite a

bit of computation. Maple has built-in capabilities that ease the computational burden and

allow us to focus on the meaning of the approximate solution.

Solved Example 1:

Solve the initial value problem y '' - x y ' + 4 y ; y (0) = 1, y '(0) = 0.

Solution:

First we note that the equation does not have constant coefficients. Next we see that x = 0 is an ordinary point of the equation. Existence and uniqueness theory guarantees that this IVP has a unique solution. The theory of series solutions also guarantees that the solution has a Maclaurin series expansion that represents the solution at every value of x . To control the size of the degree of the polynomial approximation, we invoke the Order command, first asking to see all the terms of the series up to order 3 and then up to order 10. We use the series option in dsolve :

> with(DEtools):

> Order:=3:
de:=diff(y(x),x$2)-x*diff(y(x),x)+4*y(x);

> dsolve({de,y(0)=1,D(y)(0)=0}, y(x), type=series);

> Order:=10:

> dsolve({de,y(0)=1,D(y)(0)=0}, y(x), type=series);

>

> Order:=20:

> dsolve({de,y(0)=1,D(y)(0)=0}, y(x), type=series);

It seems as if the Maclaurin series for the solution is just the polynomial . We can check this and see that we have been lucky enough to find a simple closed form solution in the form of a fourth degree polynomial.

Solved Example 2:

Solve the equation near the point x = 0.

Solution:

The only singular points are and , so that x = 0 is an ordinary point. Therefore we can represent the general solution of our equation as a Maclaurin series with radius of convergence at least . Let's get an idea of the form of the series solution:

> Order:=10:

> de2:=(5*x^2-2)*diff(y(x),x$2)+15*x*diff(y(x),x)+5*y(x)=0;

> dsolve(de2,y(x),type=series);

>

> convert(rhs(%), polynom);

Solved Example 3:

Use a series method to find the general solution of , x > 0, near the point x = 0.

Solution :

First we note that x = 0 is a regular singular point . Although a regular power series will not work, we could try to find solutions in the form of a Frobenius series , where the number r satifies a certain indicial equation. Maple handles all this with ease: