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 z and t satisfy the differential equation d²r _dr +5- + 2x = 0, r= 2,- dt dr = 0 when t = 0. 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.
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 z and t satisfy the differential equation d²r _dr +5- + 2x = 0, r= 2,- dt dr = 0 when t = 0. 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.
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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Transcribed Image Text:1. A function h(x) 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
d?r
7
dt2
dr
dr
+ 5- + 2x = 0, x = 2,
dt
= 0 when t = 0.
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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