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

Sydney Stanley 2022-05-02 Answered
$\left(1-{x}^{2}\right)y{}^{″}-x{y}^{\prime }+{m}^{2}y=0$, m is a constant. I have to show that $m\in \mathbb{Z}⇒deg\left(y\right)=m$ such that y is a solution of the ode.
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Yaretzi Odom
Step 1
Note that you have the following, which is valid for $n\ge 0$:
${a}_{n+2}=\frac{\left(n+m\right)\left(n-m\right)}{\left(n+2\right)\left(n+1\right)}{a}_{n}$
Step 2
Let $m\in \mathbb{Z}$ be any. If m is even, take , (so ). Therefore all the coefficients of the form ${a}_{2k+1}$ are zero (because of the formula above). Also, as m is an integer, ${a}_{m+2}=0$, and as it is even for all k, ${a}_{m+2k}=0$. So, .
Otherwise, if m is odd, let . Arguing in a similar way, you conclude that deg $y=m$.