Q. The solution of the differential equation $\left( x^{2} - yx^{2} \right)\frac{dy}{dx} + y^{2} + xy^{2} = 0$ is
A) $\log\left( \frac{x}{y} \right) = \frac{1}{x} + \frac{1}{y} + c$
B) $\log\left( \frac{y}{x} \right) = \frac{1}{x} + \frac{1}{y} + c$
C) $\log{(xy)} = \frac{1}{x} + \frac{1}{y} + c$
D) $\log{(xy)} + \frac{1}{x} + \frac{1}{y} = c$
Explanation
(a)
$\$The given differential equation can be rewritten as
$\Rightarrow$ $\left( \frac{1}{y^{2}} - \frac{1}{y} \right)dy = - \left( \frac{1}{x^{2}} + \frac{1}{x} \right)dx$
$\Rightarrow$ $- \frac{1}{y} - \log y = - \left( - \frac{1}{x} + \log x \right) + c$ [integrating]
$\Rightarrow$ $\log\left( \frac{x}{y} \right) = \frac{1}{x} + \frac{1}{y} + c$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.