# Second order ODE help

• October 19th 2009, 01:22 PM
mortalapeman
Second order ODE help
Determine the general solution of:

$2x^{2}y''+3xy'-3y=0$

My teacher has a bad habit of teaching us things that aren't in the book, or atleast not in the chapters we are suposse to be in, and i can't find any examples on how to do this kind of problem without an initial solution. There is an example in the book like that and it says we'll determine how to find this solution in later chapters but i can't figure out which chapter it is.

If i could just get this one worked or atleast started as an example, i can figure out the rest i'm sure, because i just don't really know how to tackle it like this.

Any help would be appreciated, thanks!
• October 19th 2009, 01:55 PM
Jester
Quote:

Originally Posted by mortalapeman
Determine the general solution of:

$2x^{2}y''+3xy'-3y=0$

My teacher has a bad habit of teaching us things that aren't in the book, or atleast not in the chapters we are suposse to be in, and i can't find any examples on how to do this kind of problem without an initial solution. There is an example in the book like that and it says we'll determine how to find this solution in later chapters but i can't figure out which chapter it is.

If i could just get this one worked or atleast started as an example, i can figure out the rest i'm sure, because i just don't really know how to tackle it like this.

Any help would be appreciated, thanks!

If we look for solutions of the form $y = x^m$, then the characteristic equation is

$
2m(m-1) + 3m - 3 = 0 \;\; \text{or }\;\;2m^2+m-3=0
$

This factors $(2m + 3)(m - 1) = 0$ so m = 1 and m = -3/2.

The two solution are $y_1 = x,\;\;y_2 = x^{-3/2}$ and the general solution $y = c_1 x + c_2 x^{-3/2}$.