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


  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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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 centre 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 2𝑦﷮ 𝑎﷮2﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯= −2𝑥﷮ 𝑏﷮2﷯﷯ 𝑦﷮ 𝑎﷮2﷯﷯ . 𝑑𝑦﷮𝑑𝑥﷯= −𝑥﷮ 𝑏﷮2﷯﷯ 𝑦﷮𝑥﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯= − 𝑎﷮2﷯﷮ 𝑏﷮2﷯﷯ 𝑦﷮𝑥﷯﷯ y’ = − 𝑎﷮2﷯﷮ 𝑏﷮2﷯﷯ Again differentiating both sides w.r.t. 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 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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