Question

If $y=\frac{x}{\ell n|cx|}$ (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. $\frac{y^2}{x^2}$
• D. $-\frac{y^2}{x^2}$

Answer 1

Answer: (D)

$y=\frac{x}{\ell n|cx|}$
$\frac{dy}{dx}=\frac{y}{x}+\varphi \left(\frac{x}{y}\right)$
$\Rightarrow \frac{\ell n|cx|-1}{(\ell n|cx|{)}^2}=\frac{1}{\ell n|cx|}+\varphi \left(\frac{x}{y}\right)$
$\Rightarrow \varphi \left(\frac{x}{y}\right)=\frac{-1}{(\ell n|cx|{)}^2}=\frac{-y^2}{x^2}$