Variables and satisfy the differential equation

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Variables

x

and

y

satisfy the differential equation

\frac{dy}{dx} = x + 2y

, and

y(0) = A

. Write

y = \sum_{k=0}^{\infty} a_k x^k

. (1) Express

a_1

and

a_2

in terms of

A

and show that, for

k \ge 2

a_{k+1} = \frac{2}{k+1} a_k

. (2) Hence find the Maclaurin series expansion of

y