Check sibling questions

Ex 9.3, 8 - Family of ellipses having foci on y-axis, center

Ex 9.3, 8 - Chapter 9 Class 12 Differential Equations - Part 2
Ex 9.3, 8 - Chapter 9 Class 12 Differential Equations - Part 3


Transcript

Ex 9.3, 8 Form the differential equation of the family of ellipses having foci on 𝑦−𝑎𝑥𝑖𝑠 and center at origin. Equation of ellipse having center at origin (0, 0) & foci on y-axis is 𝑥^2/𝑏^2 +𝑦^2/𝑎^2 =1 ∴ Differentiating Both Sides w.r.t. 𝑥 𝑑/𝑑𝑥 [𝑥^2/𝑏^2 +𝑦^2/𝑎^2 ] = (𝑑(1))/𝑑𝑥 1/𝑏^2 [2𝑥]+1/𝑎^2 [2𝑦] 𝑑𝑦/𝑑𝑥=0 2𝑥/𝑏^2 +2𝑦/𝑎^2 . 𝑑𝑦/𝑑𝑥=0 Since it has two variables, we will differentiate twice 2𝑦/𝑎^2 𝑦′=(−2𝑥)/𝑏^2 𝑦/𝑎^2 𝑦′=(−𝑥)/𝑏^2 (𝑦/𝑥)𝑦′=(−𝑎^2)/〖 𝑏〗^2 (𝑦𝑦^′)/𝑥 = (−𝑎^2)/𝑏^2 Again differentiating both sides w.r.t. x ((𝑦𝑦^′ )^′ 𝑥 − (𝑑𝑥/𝑑𝑥)(𝑦𝑦^′ ))/𝑥^2 =0 (𝑦𝑦^′ )^′ 𝑥 − (1)(𝑦𝑦^′ )=𝟎×𝒙^𝟐 (𝑦𝑦^′ )^′ 𝑥 −𝑦𝑦^′=𝟎 (Using Quotient rule and Diff. of constant is 0) (𝒚𝒚^′ )^′ 𝑥 −𝑦𝑦^′=0 (𝒚^′ 𝒚^′+𝒚𝒚′′)𝑥 −𝑦𝑦^′=0 (〖𝑦^′〗^2+𝑦𝑦′′)𝑥 −𝑦𝑦^′=0 𝑥〖𝑦^′〗^2+𝑥𝑦𝑦^′′−𝑦𝑦^′=0 𝒙𝒚𝒚^′′+𝒙〖𝒚^′〗^𝟐−𝒚𝒚^′=𝟎 (Using Product rule)

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.