Finding the Second Solution to a Homogeneous Second Order DE

[maxsthekat]

Homework Statement

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

 

[Office_Shredder]

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!)

 

[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?