2. Consider the equation of simple harmonic motion considered in Example sheet 14: d²x + w²x = 0, dt² where 2 is a positive parameter. 2 (1) (a) Use the substitution y = d to write a first-order system of differential equations equiv- alent to (1) and find the solution (x, y) to the resulting system. [Hint: to derive the system, check section 3.1 of the lecture notes] (b) Assuming that the initial condition for (1) is x = x0 and dr = 0 at t=0, transform this into an appropriate initial condition for the system you have just obtained, and solve the resulting initial value problem. (c) Consider the forced harmonic motion d²x + w² x = et. dt2 Write the equivalent system of first order linear equations and find its general solution.
2. Consider the equation of simple harmonic motion considered in Example sheet 14: d²x + w²x = 0, dt² where 2 is a positive parameter. 2 (1) (a) Use the substitution y = d to write a first-order system of differential equations equiv- alent to (1) and find the solution (x, y) to the resulting system. [Hint: to derive the system, check section 3.1 of the lecture notes] (b) Assuming that the initial condition for (1) is x = x0 and dr = 0 at t=0, transform this into an appropriate initial condition for the system you have just obtained, and solve the resulting initial value problem. (c) Consider the forced harmonic motion d²x + w² x = et. dt2 Write the equivalent system of first order linear equations and find its general solution.
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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Question
![2. Consider the equation of simple harmonic motion considered in Example sheet 14:
d²x
+ w²x = 0,
dt²
where 2 is a positive parameter.
2
(1)
(a) Use the substitution y = d to write a first-order system of differential equations equiv-
alent to (1) and find the solution (x, y) to the resulting system. [Hint: to derive the system,
check section 3.1 of the lecture notes]
(b) Assuming that the initial condition for (1) is x = x0 and dr = 0 at t=0, transform this
into an appropriate initial condition for the system you have just obtained, and solve the
resulting initial value problem.
(c) Consider the forced harmonic motion
d²x
+ w² x = et.
dt2
Write the equivalent system of first order linear equations and find its general solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb7f27c73-756c-4a47-8ea7-79b047cfb8fe%2F32e8c73c-c7bd-4db6-ad90-e3d026944999%2Fkzvfz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Consider the equation of simple harmonic motion considered in Example sheet 14:
d²x
+ w²x = 0,
dt²
where 2 is a positive parameter.
2
(1)
(a) Use the substitution y = d to write a first-order system of differential equations equiv-
alent to (1) and find the solution (x, y) to the resulting system. [Hint: to derive the system,
check section 3.1 of the lecture notes]
(b) Assuming that the initial condition for (1) is x = x0 and dr = 0 at t=0, transform this
into an appropriate initial condition for the system you have just obtained, and solve the
resulting initial value problem.
(c) Consider the forced harmonic motion
d²x
+ w² x = et.
dt2
Write the equivalent system of first order linear equations and find its general solution.
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