System of first order ODE

**LHeiner** · November 20th 2010, 06:55 AM

I should state all linear subspaces for which the solutions for t->inf or t->-inf converge against x=0.

**Ackbeet** · November 20th 2010, 11:15 AM

I think I have an idea what you mean. Let's take $t\to\infty.$ Now, we don't know the signs of $a$ and $b,$ 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 $t\to\infty,$ to show you what's going on. We have

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

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

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

Does that help?

**LHeiner** · November 21st 2010, 01:04 AM

yes thank you very much, i thought of that already!

**Ackbeet** · November 22nd 2010, 05:21 AM

So you've got the final answer, then?

**LHeiner** · November 27th 2010, 12:27 AM

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

**Ackbeet** · November 29th 2010, 05:12 AM

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.

**Danny** · November 29th 2010, 06:02 AM

I know a bit about Maple. What exactly are you trying to plot?

Thread: System of first order ODE

LinkBack

Thread Tools

Display

Similar Math Help Forum Discussions

First order nonlinear system

first order system

[SOLVED] Re-writing higher order spatial derivatives as lower order system

Formulating a second-order ordinary differential equation as a first-order system?

System of second order ODE

Search Tags