Question

If differential equation of family of curves $y\ln |cx|=x$, where c is an arbitrary constant, is $y^{\prime }=\frac{y}{x}+\varphi \left(\frac{x}{y}\right)$, for some function $\varphi$, then $\varphi \left(2\right)$ is equal to?

Answer 1

$\frac{dy}{dx}\ln |cx|+\frac{cy}{cx}=1$

$\frac{dy}{dx}\frac{x}{y}=1-\frac{y}{x}$

$\frac{dy}{dx}=\frac{y}{x}-\frac{y^2}{x^2}$

For $\varphi \left(2\right)$, we get $-\frac{1}{4}$