# Second order homogeneous DE

1. Aug 3, 2009

### maxsthekat

1. The problem statement, all variables and given/known data
4xy'' + 2y' + y = 0

2. The attempt at a solution

In class, we were given that y1 = c1Cos($$\sqrt{}x$$). We then used reduction of order to figure out the other solution

Yet, I've been trying to figure out, is how do you get y1 in the first place? To me, it doesn't seem like a Cauchy-Euler equation, I don't think I can apply annihilators to it (since it's homogeneous), and since the coefficients aren't constant, it doesn't look as if I can apply variation of parameters.

Can anyone point me in the right direction?

Thanks!!

-Max

2. Aug 3, 2009

### Office_Shredder

Staff Emeritus
If you had the method to find y1 systematically, they wouldn't give it to you. Your best bet sometimes is just guess and check (best bet doesn't mean good bet!)

3. Aug 3, 2009

### maxsthekat

How would you even begin to know to try that with the square root of x as an argument of cosine? ...Do these sorts of equations often pop up in DE? If so, is numerical approximation the rule of the land?