Question

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

$log\,c+log|x|=\frac{x}{y}$
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 $\phi \left(\frac{x}{y}\right)$
$=-\frac{y^{2}}{x^{2}}$