MATH215 December 2011
• Q1 (a) • Q1 (b) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) • Q6 (c) • Q7 (a) • Q7 (b) • Q7 (c) • Q7 (d) •
Question 04 (b)
Suppose the motion of a spring-mass system is described by the following differential equation
(b) Solve the following equation for u(t) in explicit form.
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!
What are the particular and homogeneous solutions?
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Notice that the ODE
has a nonzero right-hand side. We will begin by solving the homogeneous equation
By making the ansatz we arrive at
with roots . Complex roots of the form lead to solutions of the form
Therefore the homogeneous solution is
The right-hand side is sinusoidal with argument . We need to think of which functions (and their derivatives) that could lead to this possible right-hand side. The only such possible functions are
and so, we will find a particular solution of the form
Taking this particular solution and plugging it in to the ODE, we have:
In comparing the coefficients of and , we find that
The general solution to the ODE will be , giving
The initial conditions tell us
From this, we have,
Therefore we get that
The solution is thus