(1-x^2)y''-xy'+m^2y=0, m is a constant. I have to show that

Sydney Stanley 2022-05-02 Answered
(1x2)yxy+m2y=0, m is a constant. I have to show that mZdeg(y)=m such that y is a solution of the ode.
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Answers (1)

Yaretzi Odom
Answered 2022-05-03 Author has 16 answers
Step 1
Note that you have the following, which is valid for n0:
an+2=(n+m)(nm)(n+2)(n+1)an
Step 2
Let mZ be any. If m is even, take y(0)=1 and y(0)=0, (so a0=1 and a1=0). Therefore all the coefficients of the form a2k+1 are zero (because of the formula above). Also, as m is an integer, am+2=0, and as it is even for all k, am+2k=0. So, deg y=m.
Otherwise, if m is odd, let y(0)=0 and y(0)=1. Arguing in a similar way, you conclude that deg y=m.
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