Question

The differential equation representing the family of ellipses 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. $xyy''+x (y')^2 - yy' = 0$
• B. $x+yy'' = 0$
• C. $xyy' + y^2 - 9 = 0$
• D. $xyy' - y^2 + 9 = 0$

Answer 1

Answer: (D)

$xyy' - y^2 + 9 = 0$

Answer 2

We know that general equation of ellipse is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$
And passes through the point $(0,3)$
$\Rightarrow \frac{x^{2}}{a^{2}}+\frac{y^{2}}{9}=1$
Now differentiate the E (1) with respect to $x$, we get
$\frac{2 x}{a^{2}}+\frac{2 y}{9} y^{\prime}=0$
$\Rightarrow \frac{x}{a^{2}}=\frac{-y}{9} y'$
$\Rightarrow \frac{1}{a^{2}}=\frac{-y}{9 x} y'$
From E (1) and E (2), differential equation is
$\frac{-x y}{9} y'+\frac{y'}{9}=1$
$x y y'-y^{2}+9=0$