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

Answer 1

General equation of ellipse is given by x²/a² + y²/b² = 1 ..(i)

 

It passes through (0, 3)

=> 0 + 9/b² = 1

=> b² = 9

 

so(i) becomes

=> x²/a² + y²/9 = 1

 

Differentiate w.r.t.x

2x/a² + 2yy’/9 = 0

yy’/9 = -x/a²

y’ = -(x/y)(9/a²)

(y/x)y’ = -9/a²

 

Again differentiate w.r.t.x

(y/x)y’’ + [(xy’ – y)/x²]y’ = 0

=> xyy’’ + x(y’)² – yy’ = 0

 

 

Therefore,The differential equation representing the familyxyy’’ + x(y’)² – yy’ = 0 of ellipse is