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

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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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﷯﷯ . 𝑑𝑦﷮𝑑𝑥﷯= 2𝑥﷮ 𝑎﷮2﷯﷯ 𝑦﷮ 𝑏﷮2﷯﷯ . 𝑑𝑦﷮𝑑𝑥﷯= 𝑥﷮ 𝑎﷮2﷯﷯ 𝑦﷮𝑥﷯ . 𝑑𝑦﷮𝑑𝑥﷯= 𝑏﷮2﷯﷮ 𝑎﷮2﷯﷯ y’ 𝑦﷮𝑥﷯ = 𝑏﷮2﷯﷮ 𝑎﷮2﷯﷯ Again differentiating both sides 𝑑𝑦﷮𝑑𝑥﷯ . 𝑥 − 𝑦 . 𝑑𝑥﷮𝑑𝑥﷯﷯﷮ 𝑥﷮2﷯﷯𝑦′+ 𝑦﷮𝑥﷯ . 𝑦′′ =0 𝑥 𝑑𝑦﷮𝑑𝑥﷯ − 𝑦 . 1﷯﷮ 𝑥﷮2﷯﷯𝑦′+ 𝑦﷮𝑥﷯𝑦" =0 𝑥 𝑑𝑦﷮𝑑𝑥﷯ − 𝑦 . 1﷯﷮ 𝑥﷮2﷯﷯𝑦′+ 𝑦﷮𝑥﷯𝑦" =0 𝑥 𝑦’−𝑦﷯ 𝑦’ + 𝑥𝑦 𝑦"﷮ 𝑥﷮2﷯﷯ =0 𝑥 𝑦’ −𝑦﷯ 𝑦’ + 𝑥𝑦 𝑦"=0 𝑥 𝑦﷮′﷯﷯﷮2﷯−𝑦 𝑦﷮′﷯﷯+𝑥𝑦 𝑦﷮′′﷯﷯=0 𝒙𝒚 . 𝒚﷮′′﷯+𝒙 𝒚﷮′﷯﷯﷮𝟐﷯−𝒚 𝒚﷮′﷯=𝟎

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