Find two linearly independent solutions to the equation y"-2xy'+2y=0 in the form of a power series.

We want a solution in the form

with derivatives

Substituting and its derivatives into the ODE,

gives

Shift the index on the second sum to have it start at :

and take the first term out of the other two sums. Then we can consolidate the sums into one that starts at :

and so the coefficients in the series solution are given by the recurrence,

or more simply, for ,

Note the dependency between every other coefficient. Consider the two cases,

and so on, with the general pattern

and we would see that for all .

So we have

so that one solution is

and the other is

I've attached a plot of the exact and series solutions below with , , and to demonstrate that the series solution converges to the exact one.

why can't you?

step-by-step explanation:

30

step-by-step explanation:

idk

i hope i