I need some help with this problem, I've been working on it for a while and can't seem to come up with anything!!

Q: The functions x(t) and y(t) satisfy the predator-prey equations:

x'=x-2xy
y'=-3y+xy

a) show that the solution trajectories in x,y phase space are given by

ln(y(x^3))=x+2y+constant

b) determine the critical points of the system. solve the linear comparison system corresponding to each critical point, i.e. determine a relationship between x and y (or between u and v for translated critical points). determine the type and stability of each critical point, and sketch the trajectories in the vicinity of each.

so yeah, i'm rather clueless, any help would be appreciated, thanks

2. Originally Posted by thatgirlrocks
I need some help with this problem, I've been working on it for a while and can't seem to come up with anything!!

Q: The functions x(t) and y(t) satisfy the predator-prey equations:

x'=x-2xy
y'=-3y+xy

a) show that the solution trajectories in x,y phase space are given by

ln(y(x^3))=x+2y+constant

b) determine the critical points of the system. solve the linear comparison system corresponding to each critical point, i.e. determine a relationship between x and y (or between u and v for translated critical points). determine the type and stability of each critical point, and sketch the trajectories in the vicinity of each.

so yeah, i'm rather clueless, any help would be appreciated, thanks
a) solve the DE :
$\frac{~dy}{~dx} = \frac{-3y+xy}{x-2xy}$
or
$\frac{1-2y}{y}~dy = \frac{x-3}{x}~dx$

b) the critical points are the solution to the system
$x-2xy = 0$
$-3y+xy = 0$