series soultions to differential equations

• Apr 27th 2008, 05:34 PM
mobius2000
series soultions to differential equations
what are the requirements of a differential equation for its solution to be a power series? the function has to be analytic; what exactly does this mean? does the right hand side always have to equal zero as well?

thanks
• Apr 27th 2008, 05:38 PM
Mathstud28
Quote:

Originally Posted by mobius2000
what are the requirements of a differential equation for its solution to be a power series? the function has to be analytic; what exactly does this mean? does the right hand side always have to equal zero as well?

thanks

Analytic just means it can be locally defined by a power series..
• Apr 27th 2008, 06:03 PM
ThePerfectHacker
Quote:

Originally Posted by mobius2000
what are the requirements of a differential equation for its solution to be a power series? the function has to be analytic; what exactly does this mean? does the right hand side always have to equal zero as well?

The simple version says that if you have $y''+p(x)y'+q(x)y=0$ where $p(x),q(x)$ are continous on an interval $I$ then there exists an analytic solution $y$ to this differencial equation.