Reduction of order for (1-x²)y''+2xy'=0 and 4x²y''=y

[bathtub2007]

1. Problem Statement

Okay so I know these appear to be simple but for some reason I am having trouble finding the methods by which to solve them.

Problem 1:

This problem must be solved by reduction of order method and cannot use the y2 formula.

(1 - x^2)y'' + 2xy' = 0; y1 = 1


Problem 2:

This problem must be solved by reduction of order method and cannot use the y2 formula.

4x^2y'' + y = 0; y1 = [x^(1/2)]ln|x|

Homework Equations

y2 / y1 = u (x)

y2 = u (x) * y1 (x)

y' = u'x + u

y'' = u''x +2u'

The Attempt at a Solution

The general form for the problem is:

I have been trying to use Paul's Notes (http://tutorial.math.lamar.edu/Classes/DE/ReductionofOrder.aspx) to help me along the way but I am unable to follow for the lack of the teacher not exactly teaching the problems.


Any help getting towards the right direction would be much appreciated. Thank you.

 

[rock.freak667]

For the first one...try putting w=y' such that w'=y''

Now you have a first order DE.

For the second one, if y=y1 is a solution, then y=v*y1 is another solution.