Science:Math Exam Resources/Courses/MATH215/December 2013/Question 03 (b)
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Question 03 (b) 

Suppose the displacement x(t) of a damped massspring system subject to sinusoidal forcing of amplitude is modelled by:
(b) Now take , and find the steady periodic solution (the part of the solution x(t) which remains as ). 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

The general solution is a sum of a homogeneous solution and a particular solution. What happens to the homogeneous solution as ? What solution to you need to find? 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We seek a particular solution since the homogeneous solution will decay to 0 as . As , we will look for a particular solution of form where satisfies the ODE in complex form: so that which upon cancelling the terms and simplification becomes Solving for A we obtain
Now that we have A, we can find x by using A and Euler's identity: The steady solution is . 