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

$$\frac{dx(t)}{dt}=x'=x(t)(-a+ y(t) b)\tag{1}$$
$$\frac{dy(t)}{dt}=y'=y(t)(c- x(t) d)\tag{2}$$
(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
$$B(y,y_0)=(d/c) \exp((1/c)A(y,y_0))$$
