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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.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.5: Nonlinear Systems Of Differential Equations
Problem 1YT: YOUR TURN Consider the system of differential equations dx1dt=x1x23x1dx2dt=3x1x26x2 a. Find all...
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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.
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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