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

MATH215 December 2013

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Question 03 (a)
Suppose the displacement x(t) of a damped mass-spring system subject to sinusoidal forcing of amplitude $F_{0}$ is modelled by: $x''+2x'+5x=F_{0}\sin(2t),\quad x(0)=0,x'(0)=1$. (a) Find and sketch the solution x(t) when $F_{0}=0$ (no forcing).

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
If $F_{0}=0$, the equation is homogeneous. Can you find the homogeneous solution?

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We have a homogeneous ODE $\displaystyle x''+2x'+5x=0$ with x(0) = 0, x'(0) = 1. The roots of the corresponding characteristic polynomial $\lambda ^{2}+2\lambda +5=0$ are $\lambda ={\frac {-2\pm {\sqrt {2^{2}-4(1)(5)}}}{2}}={\frac {-2\pm {\sqrt {-16}}}{2}}={\frac {-2\pm 4i}{2}}=-1\pm 2i.$ The roots are complex, and we have a general solution of $\displaystyle x(t)=C_{1}e^{-t}\sin(2t)+C_{2}e^{-t}\cos(2t)$ We can use the initial conditions to find the constants. $x(0)=0+C_{2}=0$ implies $C_{2}=0$. Next, use the fact that the second term vanishes to calculate the first derivative of the solution. $\displaystyle x'(t)=-C_{1}e^{-t}\sin(2t)+2C_{1}e^{-t}\cos(2t)=2C_{1}$ Since x'(0) = 1 this implies $C_{1}=1/2$. The final solution is therefore $x(t)={\frac {1}{2}}e^{-t}\sin(2t).$ A plot of the solution is given below. Note that the solution is bounded between $x(t)=\pm {\frac {1}{2}}e^{-t}$ so there is an envelope. The solution must fit between the upper and lower envelope functions. At $t=0$, $x=0$, and the solution is increasing at $t=0$ (the derivative is positive).