Mass-spring systems; please help!

[heartilly89]

Hi! My differential equations professor is one of those people who is really just entirely too smart to be teaching, and with finals coming up, I'm starting to panic. I have 2 problems I'd like to know how to do, and anyone who can explain in layman's terms how to work these problems would have my eternal gratitude!

I do know the basics, I can solve a first- or second-order homogeneous D.E. most of the time, and I have managed to toil through a few of the higher level ones (with much blood, sweat, and tears), but these two are barely covered in the textbook and my professor didn't work a single example for either on the board; his lectures quote the text almost word for word, but he skips most of the examples.

1. Suppose that a mass-spring system is modeled by the differential equation y''+by'+25y=0.
a) What value of b corresponds to critical damping?
b) Find a general solution of the differential equation when b is this special value.

My textbook has 2 paragraphs on critical damping and no direction on what to do with the vague information therein. Help? I know what damping is, and critical damping is when the mass-spring system is unable to move because the damping force and some other force are equal. I have no idea how to figure out what it is, though, and I can't find a correct general solution until I have that...

2. Suppose that a mass-spring system is modeled by the differential equation y''+2y'+26y=82cos(4t). Find the steady state solution that represents the long time behavior of any solution to this differential equation.

My textbook has one paragraph and zero examples on how to deal with "steady state" solutions, and I can't find anything online that clearly and concisely explains how to do it. I THINK it has something to do with limits, but that's the extent of my knowledge on the subject.

Thanks in advance for your help!

 

[HallsofIvy]

Hi! My differential equations professor is one of those people who is really just entirely too smart to be teaching, and with finals coming up, I'm starting to panic. I have 2 problems I'd like to know how to do, and anyone who can explain in layman's terms how to work these problems would have my eternal gratitude!

You are aware, aren't you, that many of the people here, and very likely those that give the best responses are teachers? It is not very politic to begin a request for help by implying that a person who is a teacher must not be very smart! My experience has always been that the smartest people are the best teachers because they understand the material best and can give the clearest explanations.

I do know the basics, I can solve a first- or second-order homogeneous D.E. most of the time, and I have managed to toil through a few of the higher level ones (with much blood, sweat, and tears), but these two are barely covered in the textbook and my professor didn't work a single example for either on the board; his lectures quote the text almost word for word, but he skips most of the examples.

1. Suppose that a mass-spring system is modeled by the differential equation y''+by'+25y=0.
a) What value of b corresponds to critical damping?
b) Find a general solution of the differential equation when b is this special value.

My textbook has 2 paragraphs on critical damping and no direction on what to do with the vague information therein. Help? I know what damping is, and critical damping is when the mass-spring system is unable to move because the damping force and some other force are equal. I have no idea how to figure out what it is, though, and I can't find a correct general solution until I have that...

No, not "unable to move". Since you say you are able to solve "first or second-order homogeneous D.E." (I suspect you mean "linear with constant coefficients") you know you start by solving the characteristic equation, here, $\displaystyle r^2+ br+ 25= 0$, and that if you get real roots you have exponentials as solutions, if complex, sine and cosine. For what values of b are the roots real and for what values of b, complex? The boundary between those intervals gives "critical damping".

2. Suppose that a mass-spring system is modeled by the differential equation y''+2y'+26y=82cos(4t). Find the steady state solution that represents the long time behavior of any solution to this differential equation.

My textbook has one paragraph and zero examples on how to deal with "steady state" solutions, and I can't find anything online that clearly and concisely explains how to do it. I THINK it has something to do with limits, but that's the extent of my knowledge on the subject.

Thanks in advance for your help!

The solution to that equation will involve an exponential to the "-at" power for some positive a. As t increases that will rapidly go to 0. What is left is the "steady state" solution.