Question

The solution of the differential equation $ydx-(x+y^2)dy=0$ represent a family of

• A. circle
• B. parabola
• C. hyperbola
• D. ellipse

Answer 1

The correct option is B parabola

$ydx-(x+y^2)dy=0\Rightarrow \frac{dx}{dy}-\frac{x}{y}-y=0\Rightarrow \frac{dx}{dy}-\frac{x}{y}=y$
Integrating factor
$=e^{\int \frac{-1}{y}dy}=e^{-\ln y}=\frac{1}{y}$
Solution of the D.E,
$x\cdot \frac{1}{y}=\int \frac{1}{y}\cdot ydy+C\Rightarrow \frac{x}{y}=y+C\Rightarrow x=y^2+Cy$
This represent a family of parabola.