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)
Given $\frac{1 - y}{y^{2}}\ dy + \frac{1 + x}{x^{2}}\ dx = 0$
$\Rightarrow$ $\int_{}^{}\left( \frac{1}{y^{2}} - \frac{1}{y} \right)dy + \int_{}^{}\left( \frac{1}{x^{2}} + \frac{1}{x} \right)\ dx = 0$
$\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.