The differential equation of all the ellipses | vedclass

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Question

(B) The equation of an ellipse centered at the origin with axes as the coordinate axes is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. Differentiating

Vedclass · Accepted Answer

$x y y^{\prime \prime}+x y^{\prime 2}-y y^{\prime}=0$

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(B) The equation of an ellipse centered at the origin with axes as the coordinate axes is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. Differentiating with respect to $x$, we get $\frac{2x}{a^{2}}+\frac{2y y^{\prime}}{b^{2}}=0$, which simplifies to $\frac{x}{a^{2}}+\frac{y y^{\prime}}{b^{2}}=0$. This gives $\frac{b^{2}}{a^{2}} = -\frac{y y^{\prime}}{x}$. Differentiating again with respect to $x$, we get $\frac{1}{a^{2}} + \frac{1}{b^{2}}(y y^{\prime \prime} + (y^{\prime})^{2}) = 0$. Substituting $\frac{1}{a^{2}} = -\frac{y y^{\prime}}{b^{2}x}$, we have $-\frac{y y^{\prime}}{b^{2}x} + \frac{1}{b^{2}}(y y^{\prime \prime} + (y^{\prime})^{2}) = 0$. Multiplying by $b^{2}x$, we get $-y y^{\prime} + x(y y^{\prime \prime} + (y^{\prime})^{2}) = 0$. Thus, $x y y^{\prime \prime} + x(y^{\prime})^{2} - y y^{\prime} = 0$.

The differential equation of all the ellipses centered at the origin and having axes as the coordinate axes is

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