Question

If $y=\frac{x}{\ln |c x|}$ (where $c$ is an arbitrary constant) is the general solution of the differential equation $\frac{d y}{d x}=\frac{y}{x}$ $+\phi\left(\frac{x}{y}\right)$ 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)

$ \ln c +\ln | x |=\frac{ x }{ y } $
$\text { diff. w.r.t. } x , \frac{1}{ x }=\frac{ y - xy _1}{ y ^2}$
$\frac{y^2}{x}=y-x \frac{d y}{d x} $
$\frac{d y}{d x}=\frac{y}{x}-\frac{y^2}{x^2} \Rightarrow \phi\left(\frac{x}{y}\right)=-\frac{y^2}{x^2} \Rightarrow D$