# Reduction of Order for variable coefficients?

• Jan 9th 2010, 02:03 PM
Kaitosan
Reduction of Order for variable coefficients?
4x^2 y′′ + 8 x y′ − 3 y = 0 , x > 0

y(4) = 8, y′ (4) = 0

So far I've only seen characteristic solutions being solved via reduction of order but I'm not so sure about equations that include variable coefficients and in which Y1 and Y2 subsets of the general solution Y are not given.....a quick clarification on this would be much appreciated! Thanks!
• Jan 9th 2010, 03:32 PM
Chris L T521
Quote:

Originally Posted by Kaitosan
4x^2 y′′ + 8 x y′ − 3 y = 0 , x > 0

y(4) = 8, y′ (4) = 0

So far I've only seen characteristic solutions being solved via reduction of order but I'm not so sure about equations that include variable coefficients and in which Y1 and Y2 subsets of the general solution Y are not given.....a quick clarification on this would be much appreciated! Thanks!

This is a Cauchy-Euler Equation of the form $\displaystyle ax^2\frac{\,d^2y}{\,dx^2}+bx\frac{\,dy}{\,dx}+cy=0$. Supposing a solution of $\displaystyle y=x^r$ (where $\displaystyle x\neq0$), you can come up with a characteristic equation as seen here.

You can find out more about these types of equations and how to solve them in that link I gave you. Try to see if you can get anywhere with this information, and post back if you get stuck! :)
• Jan 9th 2010, 03:35 PM
The first step towards solving is assuming $\displaystyle x=e^{\xi}$ and $\displaystyle y(x)=Y(\xi)$. You can then calculate what $\displaystyle y'(x)$ and $\displaystyle y''(x)$ are in terms of $\displaystyle Y(\xi)$(by chain rule). It will reduce to a simple constant-coefficient equation for $\displaystyle Y(\xi)$. You can then plug in $\displaystyle \xi=\ln(x)$ and get your solution, $\displaystyle y(x)$.
(This is more general since it solves the equation when the roots $\displaystyle r$ are repeated or imaginary)