I don't think you're reducing the problem to two first-order systems correctly. If you look carefully, when you take the derivative of , that's really the second derivative of , which does not show up in the original DE.
Hi there, for the following equation that is in the phase plane (x,y),
<--- sorry, there was a typo originally, now correct
where
I need to prove that the solutions of the coupled, nonlinear system are given by
where C is a constant. So what I have done so far is:
So the equivalent system is:
(as stated in the question)
So from this, I tried letting
This upon solving this, I got
which seems close, but I am not sure this is the right method or not
Well, this step:
is an incorrect middle step, but you fix it in the next step. What you really mean is this:
.
Let me see. I'm not good at skipping lots of steps, so forgive me for reproducing the separation of variables:
, so
, and therefore
.
If you like, you can absorb the 2 into the .
So I got a different constant multiplying the term.
Now, I dislike factoring. I would take your target expression, and simplify:
.
If you compare this to the result we got above, you can see that the result follows by simply re-defining the constant of integration.
Sorry, another question in regard to this one:
Let's say I let C=4 and then C=5. How do I go about drawing the phase portraits for these values?
What I have done so far is for the formula
I simply let C=4
and then moving the right hand side to the left, I solved and got the equilibrium points as (0,0), (8^1/2,0), (-(8^(1/2)),0) and then how do I figure out what sort of points are they (i.e. saddle points, centres)? And then, how do I draw it??? I've tried drawing in WolframAlpha expect that only give a basic outline. how do I go about figuring out when I would get periodic solutions? Your help is much appreciated.
Well, you can always draw the phase portrait by hand as a last resort. I would think Mathematica would be able to draw a phase portrait. Try the StreamPlot command. What you need to know is the x and y components of the field at each point.