Normal Modes : Second Order Simultaneous Equations
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This question is concerned with finding the solutions of the second order simultaneous equations
where a = 38, b = -9, c = 378, d = -79
(i) Find the particular solutions to the differential equations which satisfy the initial conditions x = -10 and y = 7 at t = 0
together with the condition at t = 0..
For this part of the question give x as a function of t.
Omit the "x = " (8 marks)
Your Answer: -77(cos(4t))+ 67(cos(5t))
Comment: The particular solution required in this part is x = ""
(ii) What is y as a function of t.
Omit the "y = "
(8 marks)
Your Answer: -462(cos(4t))+ 469(cos(5t))
Comment: The particular solution required in this part is y = ""
(iii) What is the value of x at t = 0.56?
Give your answer to AT LEAST TWO PLACES OF DECIMALS. Put in this value only ie. Omit the "x = " (2 marks)
Your Answer: -15.36105271
Comment: x = -15.36105271
(iv) What is the value of y at t = 0.56?
Give your answer to AT LEAST TWO PLACES OF DECIMALS. Put in this value only ie. Omit the "y = " (2 marks)
Your Answer: -155.2952131
Comment: y = -155.2952131
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Question 1: (Normal Modes)
This question is concerned with finding the solutions of the second order simultaneous equations
where a = 38, b = - 9 , c = 378, d = - 79 . The initial conditions are
together with the conditions
Note:
In my opinion, to solve this kind of problem, using Laplace transform is the simplest and the most universal method in solving most of differential equations. Let me briefly review some properties of Laplace transform (used in this question) for you.
Denote that and . The Laplace transform of the first order derivative is
Using the Laplace transform of , the Laplace transform of the second order derivative can be found as
Let us consider the second order simultaneous differential equations in this question,
(1)
(2)
Take Laplace transform on both sides of both ...
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