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Example 6 - Family of ellipses having foci on x-axis, center

Example 6 - Chapter 9 Class 12 Differential Equations - Part 2
Example 6 - Chapter 9 Class 12 Differential Equations - Part 3

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Question 3 Form the differential equation representing the family of ellipses having foci on 𝑥−𝑎𝑥𝑖𝑠 is center at the origin. Ellipse whose foci is on x-axis & center at origin is 𝑥^2/𝑎^2 +𝑦^2/𝑏^2 =1 Differentiating both sides w.r.t. 𝑥 𝑑/𝑑𝑥 [𝑥^2/𝑎^2 +𝑦^2/𝑏^2 ]=𝑑(1)/𝑑𝑥 1/𝑎^2 ×(〖𝑑(𝑥〗^2))/𝑑𝑥+1/𝑏^2 ×(〖𝑑(𝑦〗^2))/𝑑𝑥=0 Since it has two variables, we will differentiate twice 𝑥^2/𝑎^2 +𝑦^2/𝑏^2 =1 1/𝑎^2 ×2𝑥+1/𝑏^2 ×(2𝑦 . 𝑑𝑦/𝑑𝑥)=0 2𝑥/𝑎^2 +2𝑦/𝑏^2 𝑑𝑦/𝑑𝑥=0 2𝑦/𝑏^2 𝑑𝑦/𝑑𝑥=(−2𝑥)/〖 𝑎〗^2 𝑦/𝑏^2 𝑑𝑦/𝑑𝑥=(−𝑥)/〖 𝑎〗^2 𝑦/𝑥 𝑑𝑦/𝑑𝑥= (−𝑏^2)/〖 𝑎〗^2 𝑦/𝑥 𝑦^′= (−𝑏^2)/〖 𝑎〗^2 Again differentiating both sides 𝑑(𝑦/𝑥)/𝑑𝑥. 𝑦^′+𝑦/𝑥 (𝑑(𝑦^′))/𝑑𝑥=𝑑/𝑑𝑥 ((− 𝑏^2)/( 𝑎^2 )) [𝑑𝑦/𝑑𝑥 . 𝑥 − 𝑦 .𝑑𝑥/𝑑𝑥]/𝑥^2 𝑦^′ +𝑦/𝑥 ×𝑦′′=0 [𝑦^′ 𝑥 − 𝑦]/𝑥^2 𝑦^′ +𝑦/𝑥×𝑦′′=0 Multiplying x2 both sides 𝑥^2×[𝑦^′ 𝑥 − 𝑦]/𝑥^2 𝑦^′ +𝑥^2×𝑦/𝑥×𝑦′′=𝑥^2×0 [𝑦^′ 𝑥−𝑦] 𝑦^′+𝑥𝑦𝑦^′′=0 〖〖𝑥𝑦〗^′〗^2−𝑦𝑦^′+𝑥𝑦𝑦^′′=0 𝑥𝑦𝑦^′′+〖〖𝑥𝑦〗^′〗^2−𝑦𝑦^′=0 𝒙𝒚 (𝒅^𝟐 𝒚)/(𝒅𝒙^𝟐 ) +𝒙(𝒅𝒚/𝒅𝒙)^𝟐−𝒚 𝒅𝒚/𝒅𝒙=𝟎 is the required differential equation

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.