A function is represented for all real by the Maclaurin series with

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

Problem

A function

y

is represented for all real

x

by the Maclaurin series

\displaystyle y(x)=\sum_{n=0}^{\infty} c_n x^{n}

with

c_0=1

. The function satisfies the differential equation

(1+x^{2})\,y'(x)+2x\,y(x)=0

. (a) Write the first four nonzero terms of

y(x)

in terms of the unknown coefficients

c_1, c_2, c_3

. Then compute

y'(x)

as a power series. (b) Substitute the series for

y

and

y'

into the differential equation to obtain a recurrence relation for

c_{n}

(

n\ge 1

). (c) Use the recurrence to find

c_1, c_2,

and

c_3

explicitly. (d) Determine a closed‐form expression for

c_{2m}

and for

c_{2m+1}

(

m\ge 0

). (e) Identify the elementary function represented by

y(x)

and state its radius of convergence, giving justification.