Question

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

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