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Ex 9.3, 9 - Family of hyperbolas having foci on x-axis, center

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

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Ex 9.3, 9 Form the differential equation of the family of hyperbolas having foci on 𝑥−𝑎𝑥𝑖𝑠 and center at origin. Equation of hyperbola having foci on x-axis & center at origin (0, 0) 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 [2𝑥]−1/𝑏^2 [2𝑦 . 𝑑𝑦/𝑑𝑥]=0 2𝑦/𝑏^2 . 𝑦′=2𝑥/𝑎^2 Since it has two variables, we will differentiate twice 𝑦/𝑏^2 𝑦′=𝑥/𝑎^2 (𝑦/𝑥)𝑦′=𝑏^2/𝑎^2 (𝑦𝑦^′)/𝑥 = 𝑏^2/𝑎^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 (𝒚𝒚^′ )^′ 𝑥 −𝑦𝑦^′=0 (Using Quotient rule and Diff. of constant is 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.