# Differntial Equations

• February 26th 2006, 12:13 PM
robert123
Differntial Equations
ODE:
x' = -3x^2 - x + 2

X[0] is given.

What values will yield equilibrium. I got those values 2/3 and -1.

Q: Equilibria can either be characterized by stable, indifferent or unstable. What is applicable for the above found values as : 2/3 and -1?

Another Q : Let x[0] > 0 and if we solve it by Forward Euler, what should be the max stepsize?

Any help is appreciated!

Thanks
Robert
• February 27th 2006, 02:52 AM
CaptainBlack
Quote:

Originally Posted by robert123
ODE:
x' = -3x^2 - x + 2

X[0] is given.

What values will yield equilibrium. I got those values 2/3 and -1.

Q: Equilibria can either be characterized by stable, indifferent or unstable. What is applicable for the above found values as : 2/3 and -1?

Put $y=x-2/3$, then the first of the equilibria corresponds to
$y=0$. Now rewriting the DE in terms of $y$ gives:

$
\frac{dy}{dt}=-3y(y+\frac{5}{3})
$

And if $y$ is small:

$
\frac{dy}{dt}\approx -5y
$

So we see for small perturbations about $y=0$ the DE gives a
rate of change of $y$ in the direction restoring $y$
towards zero (+ve perturbation gives -ve rate of change
and vice versa), so $y=0$ and therefore $x=2/3$ is a stable equilibrium.

RonL
• February 27th 2006, 04:18 AM
CaptainBlack
Quote:

Originally Posted by robert123
ODE:
x' = -3x^2 - x + 2

X[0] is given.

What values will yield equilibrium. I got those values 2/3 and -1.

Q: Equilibria can either be characterized by stable, indifferent or unstable. What is applicable for the above found values as : 2/3 and -1?

Now lets look at the other equilibrium point.

$
\frac{dx}{dt}=-3(x-2/3)(x+1)
$

So let $y=x+1$, then:

$
\frac{dy}{dt}=-3(y-\frac{5}{3})y
$
,

or for small perturbations $y$ is very small and so:

$
\frac{dy}{dt}\approx 5y
$
.

So the rate of change of $y$ is in the same sence as $y$, which gives us an unstable equilibrium.

RonL
• February 27th 2006, 05:30 AM
CaptainBlack
Quote:

Originally Posted by robert123
ODE:
x' = -3x^2 - x + 2

X[0] is given.

What values will yield equilibrium. I got those values 2/3 and -1.

Q: Equilibria can either be characterized by stable, indifferent or unstable. What is applicable for the above found values as : 2/3 and -1?

Another Q : Let x[0] > 0 and if we solve it by Forward Euler, what should be the max stepsize?

The answer to this second question depend on what you want the
answer for. If we are just interested in convergence to the equilibrium
without oscillations then a step size of:

$
h_{max}=|(x_0-2/3)/ x_0'|
$

will suffice. If we want to reproduce the trajectory with reasonable
verisimilitude we would use a $h<, a typical rule of thumb that
has worked reasonably well for control problems is to use

$h=h_{max}/3$.

RonL
• February 27th 2006, 09:43 AM
robert123
Thanks!
Hey ,

Thanks alot for you great help !

Robert