# Local Solution of a Nonlinear ODE Using a Power Series Expansion

Local Solution of a Nonlinear ODE Using a Power Series Expansion

Consider the nonlinear ODE , subject to the initial conditions and (set by you).

f''(x)f(x)+f'(x)+3=0

2

f(x)

f(0)=a

f'(0)=b

The Demonstration finds approximate solutions of the ODE for near 0 in the form . You can vary the order of the expansion from 1 to 5.

x

f(x)=f(0)+f'(0)x+f''(0)+f'''(0)+…

1

2!

2

x

1

3!

3

x

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 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.

x