Differential Equation: x^2y''-xy'-3y=2x^-(3/2)

[Ric-Veda]

Homework Statement

Homework Equations

The Attempt at a Solution

I am not asking to find the answer, just wanted to know whether to use the variation of parameters or undetermined coefficients. Because this was on a test problem and I used variation of parameters instead. I know it is a Cauchy-Euler equation, but do you use the method of undetermined coefficients or variation of parameters?

 

[Eclair_de_XII]

First, you want to put your equation in standard form as such:

$y''-\frac{1}{x}y'-\frac{3}{x^2}y=2x^{-\frac{7}{2}}$.

Then observe that the terms on the left side decrease in power by an increment of one. This implies that you want solutions $y$ of a form where $y$ is a function of $x$ of some power $k$, so that you can say that $y'$ is in the $k-1$ power. Then you have $y''$ in the $k-2$ power.

For example, for $y=ax^{k}$, $y'=akx^{k-1}$, and $y''=ak(k-1)x^{k-2}$.

Plug these values in the differential equation, and you have:

$ak(k-1)x^{k-2}-\frac{1}{x}(akx^{k-1})-\frac{3}{x^2}ax^{k}=ax^{k-2}(k(k-1)-k-3)=2x^{-\frac{7}{2}}$.

Now everything is in terms of the same power. I don't know how this would work for non-homogeneous equations, though. I don't exactly know what this method is called, in any case. But this is what I would have done.

 

[ehild]

Homework Statement

Homework Equations

The Attempt at a Solution

I am not asking to find the answer, just wanted to know whether to use the variation of parameters or undetermined coefficients. Because this was on a test problem and I used variation of parameters instead. I know it is a Cauchy-Euler equation, but do you use the method of undetermined coefficients or variation of parameters?

You can use both. Show your work.

 

[pasmith]

Unless the question specifically tells you to use a particular method, any valid method should be acceptable.