Differential Equation, Frobenius method

[rchalker]

Homework Statement

Find two linearly independent power series solutions for xy" - y' + xy = 0 using the Frobenius method.

Homework Equations

The Attempt at a Solution

solving for the indicial roots I got:

-> r(r-2) = 0
r = 0, 2

for the recursion formula I got:

I'm not completely sure this is correct, but if it is, I don't know where to go from here. I have a 0 in the denominator for the first few solutions when r=0. What am I doing wrong?

Thanks.

 

[HallsofIvy]

Check your textbook. Your solutions to the indicial equation differ by an integer. That case is similar to the situation in the "Euler-Cauchy" equation when you have a double root to the characteristice equation- you have to multiply by ln(x).

 

[Metaleer]

As HallsofIvy has pointed out, the two solutions to the indicial equation differ by an integer. This means that the only guaranteed solution is obtained by using the greater value of the two solutions - to get the other linearly independent solution to your homogeneous equation, you can use reduction of order; assume that the general solution looks like

$$y(x) = A(x) y_1(x),$$​

where $$y_1(x)$$ is the solution you obtain from the greater value of the indicial equation. Differentiating this and plugging into your ODE will yield a first-order linear equation, after redefining a variable.

Hope this helps. :)

 

[rchalker]

Thanks, trying it now but I have another question.

All of the examples have only

values when finding solutions, but mine have both

and

when r=2. Should this be happening? When I try to use it these solutions to find my second solution it seems to mess everything up.

This is what I got for my first solution-

 

[Metaleer]

For some Frobenius series expansions, in addition to the indicial equation, additional transition equations may appear that allow you to determine the $$c_i$$, where $$i$$ is some number in $$\mathbb{N}$$. In this example, you should have an additional transition equation to determine $$c_1$$, and the final Frobenius series is left as a function of $$c_0$$. What you have to do is write out the the first two terms of the Frobenius expansion explicitly and then group the remaining terms into one single infinite series (by doing some index manipulation acrobatics). The first two terms give, respectively, the indicial equation (which you apparently already have) and the equation to determine $$c_1$$. Don't hesitate to ask if you get stuck somewhere. :)

 

[rchalker]

I think I got it now, thanks