Finding Practical Resonance (Need help with my setup)

[ZenPhys]

Homework Statement

This is for my differential equation project and this is the last part of the question.
We know that the R=2 and C=0.001. We are given the equation:
Q''+(R/L)Q'+(1/LC)Q=(117/L)sin(120*pi*t)
We want to find the value of L such that we maximize the amplitude of the steady state solution.

I have uploaded a document with my set-up so far and I appreciate any help at all!

 

[Mark44]

Homework Statement

This is for my differential equation project and this is the last part of the question.
We know that the R=2 and C=0.001. We are given the equation:
Q''+(R/L)Q'+(1/LC)Q=(117/L)sin(120*pi*t)
We want to find the value of L such that we maximize the amplitude of the steady state solution.

I have uploaded a document with my set-up so far and I appreciate any help at all!

Instead of carrying the variables through all of your calculations, why don't you substitute in the values for R and C that you have? It would probably make your calculations easier.

 

[ZenPhys]

Oh ya. I eventually did that. But i realized that since my external force is a sin function. I can't be equate that.

Instead, I should differentiate A^2 + B^2 such that the differentiation of that is zero. That would then give me the maximum amplitude.
Would that be correct?

 

[Mark44]

I see a couple of mistakes.
1. In your general solution the exponential terms should have t in the exponent.
2. In your particular solution, you have a factor of t⁰. Did you mean t₀. Either way, why is it present?

In the original diff. equation, it seems like it would be easier to multiply through by L, rather than carry it along in all the denominators.

In the OP, you have

We want to find the value of L such that we maximize the amplitude of the steady state solution.

What is your steady state solution? What is your transient solution? The general solution has a transient part (with the exponential terms) and a steady state part (caused by the forcing function).