# Math Help - Dynamical Systems

1. ## Dynamical Systems

I have a continuous dynamical system given by

dx/dt = x(y-z),
dy/dt = y(z-x)
dz/dt = z(x-y)

and have to show that for any initial condition
x0 , y0 , z0 such that
x0 +y0+z0= k for any constant k
the solution of the above equations will always lie in the plane
x+y+z=k
originally i tried to look at the solution, i got
(x(t),y(t),z(t) )= (e^(y-z)t , e^(z-x)t, e^(x-y)t)
but couldn't see anything to do from then onwards.
i think there's a nice and simple answer i'm just not seeing as it's only two percent of the sheet so i'm guessing sides of A4 covered in calculations are out.
cheers.

2. hint: statement: dx/dt + dy/dt + dz/dt = 0

3. Originally Posted by Skalkaz
hint: statement: dx/dt + dy/dt + dz/dt = 0
Another hint

$\frac{\dot{x}}{x} + \frac{\dot{y}}{y} + \frac{\dot{z}}{z} = 0$

where

$\dot{x} = \frac{dx}{dt}$.

4. cheers - i feel a bit foolish for not seeing that myself though.