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Differential Equations

Multiple Choice Questions

491.
Let F denotes the family of ellipses whose centre is at the origin and major axis is the y-axis. Then, equation of the family F is :

$\frac{d^{2} y}{{dx}^{2}} + \frac{dy}{dx} (x \frac{dy}{dx} - y) = 0$

$xy \frac{d^{2} y}{{dx}^{2}} + \frac{dy}{dx} (x \frac{dy}{dx} - y) = 0$

$xy \frac{d^{2} y}{{dx}^{2}} + \frac{dy}{dx} (x \frac{dy}{dx} - y) = 0$

$\frac{d^{2} y}{{dx}^{2}} - \frac{dy}{dx} (x \frac{dy}{dx} - y) = 0$

$xy \frac{d^{2} y}{{dx}^{2}} + \frac{dy}{dx} (x \frac{dy}{dx} - y) = 0$

$Equation of family of ellipse is \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 On differentiating w . r . t . x, we get \Rightarrow \frac{2 x}{a^{2}} + \frac{2 y}{b^{2}} \frac{dy}{dx} = 0 . . . (i) \Rightarrow \frac{x}{a^{2}} + \frac{y}{b^{2}} \frac{dy}{dx} = 0 . . . (ii) Again differentiating w . r . t . x, we get \frac{1}{a^{2}} + \frac{y}{b^{2}} (\frac{d^{2} y}{{dx}^{2}}) + (\frac{dy}{dx}) = 0 \Rightarrow - \frac{y}{x} \frac{dy}{dx} + y \frac{d^{2} y}{{dx}^{2}} + {(\frac{dy}{dx})}^{2} = 0 [∵ from (ii)] \Rightarrow xy \frac{d^{2} y}{{dx}^{2}} + \frac{dy}{dx} (x \frac{dy}{dx} - y) = 0$