A function h(r) is given by h(x) = r cos(r). (a) Find a simple differential equation satisfied by the function. (b) Use the differential equation to produce the general term of the McLaurin series for the function. (c) Find the first six non-zero terms of the McLaurin series for the function, using the pattern in the values of the derivatives that emerge from the re- peated differentiation. (d) Suggest an alternative way to find the McLaurin series for this function.

A function h(r) is given by h(x) = r cos(r). (a) Find a simple differential equation satisfied by the function. (b) Use the differential equation to produce the general term of the McLaurin series for the function. (c) Find the first six non-zero terms of the McLaurin series for the function, using the pattern in the values of the derivatives that emerge from the re- peated differentiation. (d) Suggest an alternative way to find the McLaurin series for this function.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

1. A function h(r) is given by
h(r) = r cos(r).
(a) Find a simple differential equation satisfied by the function.
(b) Use the differential equation to produce the general term of the McLaurin
series for the function.
(c) Find the first six non-zero terms of the McLaurin series for the function,
using the pattern in the values of the derivatives that emerge from the re-
peated differentiation.
(d) Suggest an alternative way to find the McLaurin series for this function.

2. The variables r and t satisfy the differential equation
d²r
7
+5- + 2x = 0, x= 2,
dr
dr
= 0 when t = 0.
dt
dt
(a) Set up a recurrence relation for the values of the derivatives of r at t = 0.
(b) Construct a recurrence relation for the coefficients of the McLaurin series
for r(t).
(c) Find the first four non-zero terms of the McLaurin series for r(t) and state
how these coefficients will behave as n increases.

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