Find the solution to the following differential equation: $ \frac{dy}{dx} = \frac{x - y}{xy} $

Solution 1:

We write the differential equation as

\begin{align*} xyy^\prime=x-y\tag{1} \end{align*}

and follow the receipt I.237 in the german book Differentialgleichungen, Lösungsmethoden und Lösungen I by E. Kamke.

We consider $y=y(x)$ as the independent variable and use the substitution \begin{align*} v=v(y)=\frac{1}{y-x(y)}=\left(y-x(y)\right)^{-1}\tag{2} \end{align*}

We obtain from (2) \begin{align*} v&=\frac{1}{y-x}\qquad\to\qquad x=y-\frac{1}{v}\\ v^{\prime}&=(-1)(y-x)^{-2}\left(1-x^{\prime}\right)=\left(\frac{1}{y^{\prime}}-1\right)v^2 \end{align*} From (1) we get by taking $v$: \begin{align*} \frac{1}{y^{\prime}}=\frac{xy}{x-y}=\left(y-\frac{1}{v}\right)y(-v)=y-y^2v\tag{3} \end{align*}

Putting (2) and (3) together we get \begin{align*} v^{\prime}=\left(y-y^2v-1\right)v^2 \end{align*} respectively \begin{align*} \color{blue}{v^{\prime}+y^2v^3-(y-1)v^2=0}\tag{4} \end{align*}

and observe (4) is an instance of an Abel equation of the first kind.