Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin.
The standard equation of an ellipse centered at origin with foci on y-axis is: x²/b² + y²/a² = 1
Here, a > b and foci are at (0, ±c) where c² = a² - b²
Differentiate both sides with respect to x: (2x)/b² + (2y(dy/dx))/a² = 0
Simplify: x/a² + (y(dy/dx))/b² = 0 ⇒ b²x + a²y(dy/dx) = 0
Differentiate again: b² + a²[y(d²y/dx²) + (dy/dx)²] = 0
Now we have two equations:
- b²x + a²yy' = 0
- b² + a²(yy'' + (y')²) = 0
Eliminate a and b by solving these equations
From first equation: b² = -a²yy'/x
Substitute into second equation: (-a²yy'/x) + a²(yy'' + (y')²) = 0
Divide by a²: -yy'/x + yy'' + (y')² = 0
Vertical ellipses centered at origin:
The differential equation is: xy(y'') + x(y')² - yy' = 0