Question

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

• A. $x^2/y^2$
• B. $-x^2/y^2$
• C. $y^2/x^2$
• D. $-y^2/x^2$

Answer 1

Answer: (D)

$logc+log|x|=\frac{x}{y}$
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 $\varphi \left(\frac{x}{y}\right)$
$=-\frac{y^2}{x^2}$