# System of first order ODE

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• Nov 20th 2010, 06:55 AM
LHeiner
I should state all linear subspaces for which the solutions for t->inf or t->-inf converge against x=0.
• Nov 20th 2010, 11:15 AM
Ackbeet
I think I have an idea what you mean. Let's take $\displaystyle t\to\infty.$ Now, we don't know the signs of $\displaystyle a$ and $\displaystyle b,$ so we'll have to do different cases depending on which one we have. We may be able to do something clever with the signum function. I'll do the first component for $\displaystyle t\to\infty,$ to show you what's going on. We have

$\displaystyle x(t)=e^{at}(c_{1}\sin(bt)+c_{2}\cos(bt)).$

The trig functions are not going to either blow up or go to zero. Suppose $\displaystyle a>0.$ Then the exponential blows up. Since we can't have that, the only option is if $\displaystyle c_{1}=c_{2}=0.$ Otherwise, the $\displaystyle x$ component blows up. If, on the other hand, $\displaystyle a<0,$ then it doesn't matter what $\displaystyle c_{1}$ and $\displaystyle c_{2}$ are, the $\displaystyle x$ component will go to zero.

Conversely, let's say we are looking at $\displaystyle t\to-\infty.$ The situation will be precisely the opposite of the previous case. $\displaystyle a<0$ means a blow-up unless $\displaystyle c_{1}=c_{2}=0,$ and $\displaystyle a>0$ means the component goes to zero regardless of what $\displaystyle c_{1}$ and $\displaystyle c_{2}$ are.

Does that help?
• Nov 21st 2010, 01:04 AM
LHeiner
yes thank you very much, i thought of that already!
• Nov 22nd 2010, 05:21 AM
Ackbeet
So you've got the final answer, then?
• Nov 27th 2010, 12:27 AM
LHeiner
I was wondering how the plot of this system would look like (depending on (a,b) )? But I dont know how to plot it in maple?
Has anybody an idea?

thx
• Nov 29th 2010, 05:12 AM
Ackbeet
Hmm. Your solution is a space curve in 3 dimensions, with parameter t, right? Any of those CAS's should be able to plot that, although I'm not sure how to do that in Maple. I could probably figure it out in Mathematica. Captain Black is your best bet for MATLAB. I don't know who is especially good at Maple on this forum. I would definitely consult the help manual for information on parametric plotting.
• Nov 29th 2010, 06:02 AM
Jester
I know a bit about Maple. What exactly are you trying to plot?
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