# How to use differential equations to write x(t) in terms of y and y_0? {x'=-ax+bxy, y'=cy-dxy

How to use differential equations to write $x\left(t\right)$ in terms of $y$ and ${y}_{0}$?
$\left\{\begin{array}{rcrcl}{x}^{\prime }& =& -a\phantom{\rule{thinmathspace}{0ex}}x& +& b\phantom{\rule{thinmathspace}{0ex}}xy\\ {y}^{\prime }& =& c\phantom{\rule{thinmathspace}{0ex}}y& -& d\phantom{\rule{thinmathspace}{0ex}}xy\end{array}$
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Trace Arias
$\begin{array}{}\text{(1)}& \frac{dx\left(t\right)}{dt}={x}^{\prime }=x\left(t\right)\left(-a+y\left(t\right)b\right)\end{array}$
$\begin{array}{}\text{(2)}& \frac{dy\left(t\right)}{dt}={y}^{\prime }=y\left(t\right)\left(c-x\left(t\right)d\right)\end{array}$
$\begin{array}{}\text{(3)}& \frac{dx}{dy}=\frac{x}{y}\frac{-a+yb}{c-xd}\end{array}$
or
$\begin{array}{}\text{(4)}& \frac{c-xd}{x}dx=\frac{-a+yb}{y}dy\end{array}$
$\begin{array}{}\text{(5)}& c\mathrm{log}x-xd=-a\mathrm{log}y+by+{y}_{0}:=A\left(y,{y}_{0}\right)\end{array}$
The solution to (5) can be expressed in terms of Lambert W function
$\begin{array}{}\text{(6)}& x=-\left(c/d\right)W\left(-B\left(y,{y}_{0}\right)\right)\end{array}$
where
$B\left(y,{y}_{0}\right)=\left(d/c\right)\mathrm{exp}\left(\left(1/c\right)A\left(y,{y}_{0}\right)\right)$