Hey I need help with a question regarding a prey/predator model. The question is in 2 parts (paraphrasing them). (i) if there are no predators, what is the solution (e.g will the population exponentially grow/decline). (ii) if there are no prey, what is the solution (e.g will the population exponentially grow/decline).

dx/dt = 0.3x - xy/100 <--- prey

dy/dt = 15y(1-y/15) +25xy <--- predator

My solutions.

(i) dx/dt = 0.3x - xy/100

dx/dt = 0.3x (y=0)

dx/x = 0.3 dt

integrating

ln(x) = 0.3t + c

x= ke^0.3t (k = e^c)

therefore the population is increasing exponentially.

(ii) dy/dt = 15y(1-y/15) +25xy

dy/dt = 15y(1-y/15) (x=0)

dy/dt = 15y - y^2

dy/dt = -y(-15 + y)

dy/y(-15+y) = -1 dt

integrating from this point is where I'm finding difficulty. I've attempted it, but I'm pretty sure I'm wrong.

(1/y - 1/(-15+y))dy = -1 dt

ln(y) - ln(-15+y) = -t + c

ln(y/(-15+y)) = -t + c

y/(-15+y) = ke^-t (k = e^c)

y = (ke^-t)(-15 + y)

the last line doesn't make sense to me, because I was trying to get y = something that doesn't include a 'y' in it... can someone please help me?