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