Question

The differential equation representing the family of ellipse having foci either on the x-axis or on the y-axis, centre at the origin and passing through the point $(0,3)$ is?

• A. $xy{y}^{\prime }+y^2-9=0$
• B. $x+y{y}^{\prime \prime }=0$
• C. $xy{y}^{\prime \prime }+x(y^{\prime }{)}^2-y{y}^{\prime }=0$
• D. $xy{y}^{\prime }-y^2+9=0$

Answer 1

The correct option is D $xy{y}^{\prime }-y^2+9=0$

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

it passes through $(0,3)$, so it will become

$\frac{x^2}{a^2}+\frac{y^2}{9}=1$.........(1)

differentiate w.r.t x, we get

$\frac{2x}{a^2}+\frac{2y}{9}\frac{dy}{dx}=0$

Or $\frac{1}{a^2}=\frac{-y}{9x}\frac{dy}{dx}$

Substituting the above expression in the equation (1), we get

$x^2\frac{-y}{9x}\frac{dy}{dx}+\frac{y^2}{9}=1$

Or, $x^2\frac{-y}{9x}{y}^{\prime }+\frac{y^2}{9}=1$

Thus, $-xy{y}^{\prime }+y^2=-9$

Or, $xy{y}^{\prime }-y^2+9=0$