Local Solution of a Nonlinear ODE Using a Power Series Expansion

expansion order

5

f(0)

2.

f'(0)

0.

Consider the nonlinear ODE

f''(x)f(x)+f'(x)+3

2

f(x)

=0

, subject to the initial conditions

f(0)=a

and

f'(0)=b

(set by you).

The Demonstration finds approximate solutions of the ODE for

x

near 0 in the form

f(x)=f(0)+f'(0)x+

1

2!

f''(0)

2

x

+

1

3!

f'''(0)

3

x

+…

. You can vary the order of the expansion from 1 to 5.

The green curve is the numerical solution obtained using Mathematica's built-in function NDSolve and the approximate power series expansion is drawn in red. The approximate solution (a polynomial) is shown; it agrees very well with the numerical solution for

x

near 0. The light blue pane defines the region where the absolute value of the difference between the two functions is less than 0.01.