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Form the differential equation of the family of ellipses havin

Question

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

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Text solutionVerified

Solution:

E = \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

differenciate with respect to x

E^{'} = \frac{2 x}{a ^{2}} + \frac{2 y y ^{'}}{b ^{2}} = 0

\frac{a ^{2}}{b ^{2}} = \frac{- x}{y y ^{'}} - (1)

again differentiate with respect to x

E^{''} = \frac{2}{a ^{2}} + \frac{2 y ^{' 2} + 2 y y ^{'}}{b ^{2}} = 0

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

from equation 1 and 2

\frac{x}{y y ^{'}} = \frac{1}{y ^{' 2} + 2 y y ^{'}}

x y^{' 2} + x y y^{'} = y y^{'}

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Question Text	Form the differential equation of the family of ellipses having foci on y-axis and centre at origin.
Answer Type	Text solution:1
Upvotes	150