Question

The differential equation of all the ellipses centred at the origin and have axes as the co-ordinate axes is
(A) \(y^{2}+x y^{\prime 2}-y y^{\prime}=0\)
(B) \(x y y^{\prime \prime}+x y^{\prime 2}-y y^{\prime}=0\)
(C) \(y y^{\prime \prime}+x y^{\prime 2}-x y^{\prime}=0\)
(D) \(x^{2} y^{\prime}+x y^{\prime \prime}-3 y=0\)
Where \(\mathrm{y}^{\prime} \equiv \frac{\mathrm{d} y}{\mathrm{~d} \mathrm{x}}, \mathrm{y}^{\prime \prime} \equiv \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\)

Answer 1

Ans: (B)
Hint : equation of ellipse centred at origin and have axes as the coordinate axes is
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
$$
$$
\begin{aligned}
&\Rightarrow \quad \frac{x}{a^{2}}+\frac{y y^{\prime}}{b^{2}}=0 \\
&\Rightarrow \quad \frac{-b^{2}}{a^{2}}=\frac{y y^{\prime}}{x} \\
&\Rightarrow \quad 0=\frac{x\left(y^{\prime \prime} y^{\prime}+y^{\prime 2}\right)-y y^{\prime}}{x^{2}} \\
&\therefore \quad x y^{\prime \prime} y^{\prime}+x y^{\prime 2}-y y^{\prime}=0
\end{aligned}
$$

Answer 2

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Answer 3

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