Quick question about method of Frobenius

[TroyElliott]

So, when you use the Frobenius method on a differential equation, you assume a solution Σa_k*x^(k+s). Sometimes you get more than one solution for s in the indicial equation. Is the sum of these two solutions you get from evaluating the rest of the problem with each s solution the approximation at what ever point you were investigating on the ODE or are they two separate things not to be added together?

 

[haruspex]

So, when you use the Frobenius method on a differential equation, you assume a solution Σa_k*x^(k+s). Sometimes you get more than one solution for s in the indicial equation. Is the sum of these two solutions you get from evaluating the rest of the problem with each s solution the approximation at what ever point you were investigating on the ODE or are they two separate things not to be added together?

If it's linear (which it must be to apply this method?) then necessarily a linear sum of solutions is also a solution.

 

[MidgetDwarf]

There are three cases when using the method of Frobenius:

R1>R2 and the difference is an integer

R1>R2 and the difference is not an integer

R1=R2 (repeated roots).

Note R1 and R2 denotes roots.
Tell me what happens in these 3 cases.

From here I can see what you know and don't know.