2nd order DE question :confused:

[Kevin2341]

Homework Statement

The equation for the undamped motion with no rubber band:
y" + k₁y = -10
k₁ = any number between 12 and 13

Find exact solutions using a couple different initial conditions

And then plot this phase plane using some software

The Attempt at a Solution

So I know ahead of time that my solution needs to be in a vector form in order to plot it in a phase plane (using the software that I have for this class)

Here's my attempt thus far:
I have chosen my k₁ to be 12.25 (12.25 squared is 3.5, I tried to pick nice numbers, at least as nice as a number between 12 and 13 can be)

Making y' = v
v' = -12.25y -10

Here I am getting really messed up. I know the shortcut where you can convert the y" to a λ², the y' to a λ, and the y becomes a constant, so basically you get a polynomial equation in which you can find the roots (eigenvalues) and work from there. However, in every system we worked with in this class, there was never in one condition a problem involving a constant, and I can't seem to find any help pertaining as to what I do with it.

My normal course of action here would be to find a corresponding vector A, and from there, using the eigenvalues, I can find the eigenvectors. I know that because of the value of y (being between 12 and 13), that the eigenvalues will be imaginary. This will lead to using euler's formula, and from there it's a matter of selecting initial conditions and graphing this thing, however, I can't figure out what I am supposed to do with that darn 10. Does it just add into 12.25 to become 22.25? Does it just "showup" somewhere later on?

I could really use some help

 

[SteamKing]

12.25 squared is 3.5?

It seems there is a portion of the problem statement omitted. What does your DE describe?

You have a non-homogeneous second order ODE. There is a specific procedure which must be followed in order to obtain a solution. Do you remember what this procedure is? Have you selected your initial conditions?

 

[Kevin2341]

A Mass-Spring System with a Rubber Band

1. The equation for undamped motion with no rubber band is

y" + k₁ = -10

Once you choose a value for k1 (between 12 and 13) use methods learned in sections 3.6 and 4.1 to obtain
exact solutions for a couple of different initial conditions. Use PPlane to show phase portraits for these
solutions and give a verbal description of the motion.

As for the method on solving this: We have been using the "guessing" method. In which you make an educated guess for a value of y, then take the y' and y" of that, and plug it into the equation (It usually comes out to being in a form of Ae^(lambda*t)) And then you solve for A, and plug that A value in for your educated guess giving you a Y_h

and for guessing initial conditions, I'll be quite honest in saying I have absolutely no idea where and how to pick initial conditions. I can never figure out if the y and y' values are related to each other, or if they are just arbitrary values picked. It seems a lot of the values I've seen in examples in class are usually something along the lines of:
y(0) = 2 (or some constant number)
y'(0) = 0

which leads me to think that the y' is usually representing (obviously) the derivative of the y initial.