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Differential Equations

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Series Solutions of a Second Order ODE

DIFFEQ-ENEZSE

Consider $y''+4y=0$. By plugging in $y=\sum_{n=0}^{\infty}a_nx^n$, we find two power series solutions:

$$y_1(x)=\sum_{k=0}^{\infty}\frac{(-4)^k}{(2k)!}x^{2k}$$
$$y_2(x)=\sum_{k=0}^{\infty}\frac{(-4)^k}{(2k+1)!}x^{2k+1}$$

Which of the following statements are true?

A

The first is $\cos(2x)$ and the second is $\sin(2x)$ and they have a Wronskian of them is $2$.

B

By direct computation $y_1(0)=0$ and $y_1'(0)=0$, so the Wronskian is zero regardless of $y_2(0)$ and $y_2'(0)$.

C

The radii of convergence for both solutions are $\rho=1$.

D

The Wronskian of them at $x=0$ is $1$ and they form a fundamental set.