Question

The differential equation representing the family of ellipses with centre at origin and foci on $x$-axis is given as

• A. $x{y}^{\prime }+y=0$
• B. $x^2{y}^2{\left(y^{\prime \prime }\right)}^2+{\left(x{y}^{\prime }\right)}^2+y{y}^{\prime }=0$
• C. $xy{y}^{\prime \prime }+x{\left(y^{\prime }\right)}^2-y{y}^{\prime }=0$
• D. none of these

Answer 1

Answer: (C)

Family of ellipses satisfying given conditions can be represented graphically as shown.



The equation of ellipse with centre at the origin and foci on $x$-axis is given by
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Differentiating w.r.t. $x$, we get
$\frac{2x}{a^2}+\frac{2y}{b^2}\cdot \frac{dy}{dx}=0$
$\Rightarrow \frac{y}{x}\left(\frac{dy}{dx}\right)=-\frac{b^2}{a^2}$
Again differentiating w.r.t. $x$ , we get
$\left(\frac{y}{x}\right)y^{\prime \prime }+y^{\prime }\left[\frac{y^{\prime }x-y}{x^2}\right]=0$
or $xy{y}^{\prime \prime }+x{\left(y^{\prime }\right)}^2-y{y}^{\prime }=0$