Method of undertermined coefficients

In summary, the conversation discusses the motion of an undamped mass-spring system described by a differential equation. The question of interest is for what values of w will the system exhibit resonance. The method of undertermined coefficients is suggested to find a particular solution in the case where w is not the resonant frequency. The conversation also mentions the steps to solve a second-order differential equation, which involves solving the corresponding homogeneous equation and finding a particular solution before adding them together. It is suggested to solve b) before attempting a).
  • #1
lordy12
36
0

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.
 
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  • #2


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).
 
  • #3


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.
 

Related to Method of undertermined coefficients

What is the Method of Undetermined Coefficients?

The Method of Undetermined Coefficients is a mathematical technique used to find particular solutions to nonhomogeneous linear differential equations. It involves guessing a solution based on the form of the nonhomogeneous term and then solving for the coefficients of the guess.

When is the Method of Undetermined Coefficients used?

The Method of Undetermined Coefficients is used when solving nonhomogeneous linear differential equations with constant coefficients. It is not applicable to equations with variable coefficients or non-linear equations.

How does the Method of Undetermined Coefficients work?

The method involves guessing a solution to the nonhomogeneous equation based on the form of the nonhomogeneous term. This guess is then substituted into the original equation and the coefficients are solved for. The final solution is a combination of the guess and the complementary solution.

What are the advantages of using the Method of Undetermined Coefficients?

The Method of Undetermined Coefficients is relatively simple and straightforward to use compared to other techniques for solving nonhomogeneous differential equations. It is also applicable to a wide range of equations and can often provide an exact solution without the need for further approximation.

What are the limitations of the Method of Undetermined Coefficients?

The method can only be applied to linear differential equations with constant coefficients and is not suitable for non-linear equations. It also relies on making a correct guess for the particular solution, which may not always be possible. In some cases, the method may also result in a general solution rather than an exact solution.

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