Question

From the differential equation of the family of ellipses having foci on y-axis and centre at origin.

Answer 1

The equation of the family of the ellipses having foci on the y-axis and the centre at origin is as follows:

\(\frac{x2}{b2}+\frac{y^2}{a^2}=1...(1)\)

Differentiating equation(1)with respect to x,we get:

\(\frac{2x}{b^2}+\frac{2yy}{b^2}=0\)

\(⇒\frac{x}{b^2}+\frac{yy}{a^2}=0...(2)\)

Again, differentiating with respect to we get:

\(\frac{1}{b^2}+\frac{y.y+y.y}{a^2}=0\)

\(⇒\frac{1}{b^2}+\frac{1}{a^2}(y^2+yy)=0\)

\(⇒\frac{1}{b^2}=-\frac{1}{a^2}(y^2+yy)\)

Substituting this values in equation(2),we get:

\(x[-\frac{1}{a^2}((y)2+yy)]+\frac{yy}{a^2}=0\)

\(⇒-x(y)^2-xyy+yy=0\)

\(⇒xyy+x(y)2-yy=0\)

This is the required differential equation.