A useful approach to analyzing a nonlinear equation is to study its linearized equation, which is obtained by replacing the nonlinear terms by linear approximations. For example, the nonlinear equation (8) d²0 di + sin 0 = 0, which governs the motion of a simple pendulum, has d²0 di (9) as a linearization for small 0. (The nonlinear term sin has been replaced by the linear approxi- mation 0.) + 0 = 0 A general solution to equation (8) involves Jacobi elliptic functions (see Project C), which are rather complicated, so let's try to approximate the solutions. For this purpose we consider two methods: Taylor series and linearization. (a)Derive the first six terms of the Taylor series about t = 0 of the solution to equation (8) with initial conditions (0) = π/12, 0' (0) = 0.(The Taylor series method is dis- cussed in Project D of Chapter 1 and Section 8.1.)
A useful approach to analyzing a nonlinear equation is to study its linearized equation, which is obtained by replacing the nonlinear terms by linear approximations. For example, the nonlinear equation (8) d²0 di + sin 0 = 0, which governs the motion of a simple pendulum, has d²0 di (9) as a linearization for small 0. (The nonlinear term sin has been replaced by the linear approxi- mation 0.) + 0 = 0 A general solution to equation (8) involves Jacobi elliptic functions (see Project C), which are rather complicated, so let's try to approximate the solutions. For this purpose we consider two methods: Taylor series and linearization. (a)Derive the first six terms of the Taylor series about t = 0 of the solution to equation (8) with initial conditions (0) = π/12, 0' (0) = 0.(The Taylor series method is dis- cussed in Project D of Chapter 1 and Section 8.1.)
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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Working on a project example from textbook, not for marks, just trying to understand better. Please only do a,b,c. To help with my understanding in part c, please discuss the plot and explain which approximation is better and why?
Transcribed Image Text:A useful approach to analyzing a nonlinear equation is to study its linearized equation, which is
obtained by replacing the nonlinear terms by linear approximations. For example, the nonlinear
equation
(8)
d²0
di
+ sin 0 = 0,
which governs the motion of a simple pendulum, has
d²0
di
+ 0 = 0
as a linearization for small 0. (The nonlinear term sin has been replaced by the linear approxi-
mation 0.)
A general solution to equation (8) involves Jacobi elliptic functions (see Project C), which
are rather complicated, so let's try to approximate the solutions. For this purpose we consider two
methods: Taylor series and linearization.
(a)Derive the first six terms of the Taylor series about t = 0 of the solution to equation (8)
with initial conditions (0) = π/12, 0' (0) 0.(The Taylor series method is dis-
cussed in Project D of Chapter 1 and Section 8.1.)
=
![as a linearization for small 0. (The nonlinear term sin has been replaced by the linear approxi-
mation 0.)
A general solution to equation (8) involves Jacobi elliptic functions (see Project C), which
are rather complicated, so let's try to approximate the solutions. For this purpose we consider two
methods: Taylor series and linearization.
(a)Derive the first six terms of the Taylor series about t = 0 of the solution to equation (8)
with initial conditions 0(0) = π/12, 0' (0) = 0.(The Taylor series method is dis-
cussed in Project D of Chapter 1 and Section 8.1.)
= 0.
(b) Solve equation (9) subject to the same initial conditions (0) = π/12,0' (0)
(c) On the same coordinate axes, graph the two approximations found in parts (a) and (b).
(d)Discuss the advantages and disadvantages of the Taylor series method and the lineariza-
tion method.
(e) Give a linearization for the initial value problem.
x"(t) + 0.1[1-x² (t) ]x' (t) + x(t) = 0 x(0) 0.4, x' (0) = 0,
for x small. Solve this linearized problem to obtain an approximation for the nonlinear
problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F160fd351-17ca-4577-bffb-265c7021964b%2Fc18b704a-265c-4e32-a73f-09638f17b795%2F5zkafs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:as a linearization for small 0. (The nonlinear term sin has been replaced by the linear approxi-
mation 0.)
A general solution to equation (8) involves Jacobi elliptic functions (see Project C), which
are rather complicated, so let's try to approximate the solutions. For this purpose we consider two
methods: Taylor series and linearization.
(a)Derive the first six terms of the Taylor series about t = 0 of the solution to equation (8)
with initial conditions 0(0) = π/12, 0' (0) = 0.(The Taylor series method is dis-
cussed in Project D of Chapter 1 and Section 8.1.)
= 0.
(b) Solve equation (9) subject to the same initial conditions (0) = π/12,0' (0)
(c) On the same coordinate axes, graph the two approximations found in parts (a) and (b).
(d)Discuss the advantages and disadvantages of the Taylor series method and the lineariza-
tion method.
(e) Give a linearization for the initial value problem.
x"(t) + 0.1[1-x² (t) ]x' (t) + x(t) = 0 x(0) 0.4, x' (0) = 0,
for x small. Solve this linearized problem to obtain an approximation for the nonlinear
problem.
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