Method of undertermined coefficients

[lordy12]

Homework Statement

The motion of an undamped mass-spring system is described by the differential equation 2x" + 36x = sin(wt)

a) For what values of w will the system exhibit resonance?
b) Use the method of undertermined coefficients to find a particular solution in the case where w is not the resonant frequency.





I know I have to use y = Asin(wt) but I am stuck after that.

 

[ideasrule]

Do you know how to solve a second-order differential equation? First you solve the corresponding homogeneous equation, then find a particular solution, then add the two. You may find it easier to do a) after doing b).

 

[kerimek]

I would advise approaching this problem by first understanding the concept of resonance in a mass-spring system. Resonance occurs when the driving frequency (in this case, w) matches the natural frequency of the system. This can result in large amplitude oscillations and can potentially damage the system if not properly controlled.

To answer the first question, the system will exhibit resonance when w = 6, as this is the natural frequency of the system (obtained by solving for w in the differential equation).

For the second question, the method of undertermined coefficients can be used to find a particular solution for any value of w that is not the resonant frequency. This method involves assuming a solution of the form y = Asin(wt) and plugging it into the differential equation to solve for A.

In this case, we can rewrite the differential equation as 2x" + 36x - sin(wt) = 0. Plugging in y = Asin(wt), we get:

2(-Aw^2sin(wt)) + 36(Asin(wt)) - sin(wt) = 0

Simplifying, we get:

(Aw^2 - 36A + 1)sin(wt) = 0

Since sin(wt) cannot equal 0 for all values of t, the coefficient inside the parentheses must equal 0. This gives us the equation Aw^2 - 36A + 1 = 0. We can solve for A using the quadratic formula, which gives us two possible values for A: A = 1/2 or A = 1/18.

Therefore, the particular solution for this system is y = (1/2)sin(wt) or y = (1/18)sin(wt), depending on the value chosen for A. This solution will hold for any value of w that is not the resonant frequency of the system.