d²x Consider the equation of simple harmonic motion considered in Example sheet 14: (1) + w²x = 0, dt2 where w² is a positive parameter. 2 (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 d = 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.
d²x Consider the equation of simple harmonic motion considered in Example sheet 14: (1) + w²x = 0, dt2 where w² is a positive parameter. 2 (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 d = 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 w² 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]
dt
(b) Assuming that the initial condition for (1) is x = x0 and d = 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
dt²
+w²x = et.
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%2F5af7cd69-9bae-4388-950f-d76deeea9908%2F6815435f-275e-4296-8d2c-ca0a40830c50%2F08vizrs_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 w² 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]
dt
(b) Assuming that the initial condition for (1) is x = x0 and d = 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
dt²
+w²x = et.
Write the equivalent system of first order linear equations and find its general solution.
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