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Differential Equations

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Moderate

Forms of Series Solutions

DIFFEQ-EXYSLH

Of the forms suggested for series solutions about $x_0=0$, which one of the following statements is incorrect?

A

For equation $y''+x^2y=0$, we should try $y=\sum_{n=0}^{\infty}a_nx^n$.

B

For equation $y''+x^2y=0$, there is no solution of the form $y=\sum_{n=0}^{\infty}a_nx^n$ and we should try $y=x^p\sum_{n=0}^{\infty}a_nx^n$ for some $p\in\mathbb{R}$, $a_0\neq 0$.

C

For equation $x^2y''+4xy'+(1+x)y=0$, $x=0$ is a regular singular point. There is a solution of the form $y(x)=x^p\sum_{n=0}^{\infty}a_nx^n$ for some $p\in\mathbb{R}, a_0\neq 0$.

D

For the Euler equation $x^2y''+3xy'+5y=0$, we can't find series solutions of the form $y(x)=\sum_{n=0}^{\infty}a_nx^n$ or $y(x)=x^p\sum_{n=0}^{\infty}a_nx^n$ since the roots of the equation $r(r-1)+3r+5=0$ are not real.