Stuck trying to solve a non homogeneous differential equation

[dooogle]

Homework Statement

dx/dt=x(1-2y) t(0)=0 x(t(0))=1

dy/dt=-y(1-2x), t(0)=0 y(t(0))=2

inerval of integration= [0,40]

Homework Equations

The Attempt at a Solution

i let x=p(t) so t=(p^-1)(x)

dy/dx=(∂y/∂(p^-1))∂(p^-1)/∂x

dy/dx=(-y(1-2x))/(x(1-2y))

since this equation is non homogeneous cannot make a substitution y=vx

the equation can be rearranged into the form

M(x,y)+N(x,y)dy/dx=0

where M(x,y)=y+2xy

and N(x,y)=x-2xy

but the equation is not exact as there is no function Ψ(x,y) such that

Ψ_x(x,y) =M(x,y) and Ψ_y(x,y)=N(x,y)

ive tried finding an integrating factor to make the equation exact using the formula

dμ/dx=μ((M_y-N_x)/N)

but M_y=1+2x

N_x=1-2y

((M_y-N_x)/N)=(1+2x-1-2y)/x-2xy

so

please could you tell me if i have made mistakes or point me in the direction i should be going

thank you for your time

dooogle

 

[epenguin]

In your dy/dx=(-y(1-2x))/(x(1-2y))

the variables are separable i.e. you get

(1/x - 2) dx = (-1/y + 2) dy ->

d(ln x - 2x) = d(- ln y + 2y)

Should be able to finish.

Let us know, with pics if possible.

I think these are a case of Volterra-Lotka equations.