Question

If $y=\frac{x}{\log |cx|}$ (where $c$ is an arbitrary constant) is the general solution of the differential equation
$\frac{dy}{dx}=\frac{y}{x}+\varphi \left(\frac{x}{y}\right)$, then the function $\varphi \left(\frac{x}{y}\right)$ is

• A. $\frac{x^2}{y^2}$
• B. $-\frac{x^2}{y^2}$
• C. $\frac{y}{x}$
• D. $-\frac{y^2}{x^2}$

Answer 1

The correct option is D $-\frac{y^2}{x^2}$

$\log c+\log |x|=\frac{x}{y}$
Differentiating w.r.t. $x$, $\frac{1}{x}=\frac{y-x\frac{dy}{dx}}{y^2}$
or $\frac{y^2}{x}=y-x\frac{dy}{dx}$
or $\frac{dy}{dx}=\frac{y}{x}-\frac{y^2}{x^2}$
or $\varphi \left(\frac{x}{y}\right)=-\frac{y^2}{x^2}$