- #1

metstandard16

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I have to solve this 2nd order non-linear ODE, and I'm stuck at the beginning. We have to find the equilibrium points, linearize the system, draw the phase portraits and classify the eq points, and solve it numerically.

I think I'm fine with linearizing it and classifying, as well as producing some preliminary phase portraits before I head to matlab.

The ODE:

y''-y'+y

^{2}y'+y-y

^{5}=0

I've rewritten this 2nd order ODE into (2) 1st order ODE's to start:

u=y

v=y'

u' = y' = v

v'=y'' = v-u

^{2}v-u+u

^{5}

The second equation can be factored out to = (u-1)(u+1)(u

^{3}-v+u)

My confusion lies with the fact that I have a y' in my equation and this presents a v variable subsequently. Do I have to rewrite this equation in terms of v, so it's purely with u's (for the lack of better terms)? Then find where u' = 0? None of my other examples have a (v)/y' in them so it's difficult to reference my notes. Somehow I think I have 3 eq points at (0,0) , (1,0) , & (-1,0) but I wasn't completely sure how that equation panned out.

Thanks for your help!