How to use differential equations to write $x(t)$ in terms of $y$ and $y_0$?

Question

The equations are: $$ \left\{\begin{array}{rcrcl} x' & = & \mbox{}-a\,x & + & b\,xy \\ y' & = & c\,y & - & d\,xy \end{array}\right. $$ They want me to write an equation for $x(t)$ in terms of $y$ and $y_0$ only.

How would I do this?

Answer 1

$$\frac{dx(t)}{dt}=x'=x(t)(-a+ y(t) b)\tag{1}$$ $$\frac{dy(t)}{dt}=y'=y(t)(c- x(t) d)\tag{2}$$

(1)/(2) leads to:

$$\frac{dx}{dy}=\frac{x}{y}\frac{-a+ y b}{c- x d}\tag{3}$$

or

$$\frac{c- x d}{x}dx=\frac{-a+ y b}{y}dy\tag{4}$$

Integration of (4) leads to

$$ c\log x- x d=-a\log y+b y+y_0 :=A(y,y_0)\tag{5}$$

The solution to (5) can be expressed in terms of Lambert W function

$$x= -(c/d)W(-B(y,y_0)) \tag{6}$$

where

$$B(y,y_0)=(d/c) \exp((1/c)A(y,y_0))$$

Answer 2

Hint: $\dfrac{dx}{dy} = \dfrac{x'}{y'}$.