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Differential Equations

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Series Solutions for Hermite Equations

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Consider the Hermite equation $y''-2xy'+2ny=0$ where $n$ is a positive integer.

Which one of the following statements is wrong regarding the power series solution $y=\sum_{k=0}^{\infty}a_kx^k$?

A

$x=0$ is an ordinary point so we can seek power series solutions of the form $y=\sum_{k=0}^{\infty}a_kx^k$.

B

There is a polynomial solution whose degree is $n$.

C

Every power series solution of the form indicated in the problem terminates at $x^n$, i.e., all coefficients $a_k=0$ for $k>n$.

D

There are two independent power series solutions about $x=0$, which span the solution space of the Hermite equation.