First slide

Formation of differential equations

Question

The differential equation of the ellipse with centre at origin and ends of one axes of symmetry at $(\pm 1, 0)$ is

Moderate

A

$(x^{2} - 1) y^{1} - xy = 0$
B

$(x^{2} + 1) y^{1} - xy = 0$
C

${xy}^{1} + (x^{2} + 1) y = 0$
D

$(x^{2} - 1) y^{11} + (x- 1) y^{1} = 0$

Solution

Let the ends of other axis of symmetry be $(0, \pm a)$ . Then the equation of the ellipse is $x^{2} + \frac{y^{2}}{a^{2}} = 1$
$\begin{array}{l} \to a^{2} x^{2} + y^{2} = a^{2} . . . . . . (1) \\ Dwr to x \\ \to 2 a^{2} x + 2 y y^{1} = 0 \\ \to a^{2} = \frac{- y}{x} \frac{dy}{dx} . . . . . . . (2) \\ From 1 and 2 \Rightarrow (x^{2} - 1) \frac{d y}{d x} - x y = 0 . \end{array}$

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