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Q.
The differential equation representing the family of ellipses with centre at origin and foci on $x$-axis is given as
Differential Equations
Solution:
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
$\left(\frac{y}{x}\right) y''+y'\left[\frac{y'x-y}{x^{2}}\right]=0$
or $xy\,y''+x\left(y'\right)^{2}-yy'=0$
