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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