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Differential Equations

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Moderate

Series Solutions for Nonlinear Equations

DIFFEQ-JRXHSX

Consider the equation $y'=1+x-y^2$. This is nonlinear equation but we may also pursue power series solution near $x=0$ by trying the ansatz $y=\sum_{n=0}^{\infty}a_nx^n$.

Then, $a_0, a_1, a_2,a_3$ for the solution satisfying $y(0)=1$ are equal to:

A

$0,1,\cfrac{1}{2}, -\cfrac{1}{3}$

B

$1,0, \cfrac{1}{2},-\cfrac{1}{3}$

C

$1,1, \cfrac{1}{2},-\cfrac{2}{3}$

D

$1,0,\cfrac{1}{2},-\cfrac{2}{3}$