# Second Order ODE Homogenous

1. Oct 22, 2015

### Mark Brewer

1. The problem statement, all variables and given/known data
(Reduction of order) The function y1 = x-1/2cosx is one solution to the differential equation x2y" + xy' + (x2 - 1/4) = 0. Use the method of reduction of order to find another linearly independent solution.

3. The attempt at a solution

I divided x2 to both sides to get the equation into y" + py' + qy = 0

y" + (1/x)y' + ((1 - (1/4x2)) = 0

Using Abel's method

c e-∫(p)dx

= c e-ln x
y2 = -c lnx

Am I doing this right?

2. Oct 23, 2015

### SteamKing

Staff Emeritus
Take y2 and plug it back into the original differential equation and see if it is satisfied.

3. Oct 23, 2015

### vela

Staff Emeritus
No, the problem said to use the method of reduction of order.

But first, did you type the differential equation and solution correctly? I ask because the supposed solution doesn't satisfy the given differential equation.

4. Oct 23, 2015

### HallsofIvy

Staff Emeritus
You mean x2y" + xy' + (x2 - 1/4)y = 0, don't you? As Vela pointed out, the given y1 does not satisfy the equation you gave. Perhaps it satisfies this one. I haven't checked.

So you start by looking for a solution of the form $y_2= u(x)x^{1/2}cos(x)$
Find the first and second derivatives of that and put them into the equation. If your given function really is a solution to the differential equation, then all terms involving only "u" (as opposed to u' or u'') will cancel leaving a first order equation for v= u'.