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The differential equation representing the family of ellipse having foci either on the x-axis or the y-axis centre at the origin and passing through the point (0, 3) is -
- xyy'' + y² = 9 = 0
- x + y'' = 0
- xy'' + x(y')² − yy'' = 0
- xyy' − y² + 9 = 0
Correct answer: xyy' − y² + 9 = 0
Solution
The ellipse through (0,3) with centre at origin is x^2/A + y^2/9 = 1. Differentiating: 2x/A + 2yy'/9 = 0 -> A = -9x/(yy'). Substituting back gives y^2 - xyy' = 9, i.e. xyy' - y^2 + 9 = 0.
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