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Example 6 - Family of ellipses having foci on x-axis, center - Formation of Differntial equation when general solution given


  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Example 6 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﷯﷯ 𝑑𝑥﷮2﷯﷮𝑑𝑥﷯﷯+ 1﷮ 𝑏﷮2﷯﷯ 𝑑 𝑦﷮2﷯﷮𝑑𝑥﷯﷯=0 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﷯﷯ Again differentiating both sides 𝑑 𝑦﷮𝑥﷯﷯﷮𝑑𝑥﷯. 𝑑𝑦﷮𝑑𝑥﷯+ 𝑦﷮𝑥﷯ 𝑑﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯= 𝑑﷮𝑑𝑥﷯ − 𝑏﷮2﷯﷮ 𝑎﷮2﷯﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ . 𝑥 − 𝑦 . 𝑑𝑦﷮𝑑𝑥﷯﷯﷮ 𝑥﷮2﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ + 𝑦﷮𝑥﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯=0 𝑑𝑦﷮𝑑𝑥﷯ . 𝑥 − 𝑦 ﷯ 𝑑𝑦﷮𝑑𝑥﷯ + 𝑥𝑦 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯﷮ 𝑥﷮2﷯﷯=0 𝑥 𝑑𝑦﷮𝑑𝑥﷯﷯﷮2﷯−𝑦 . 𝑑𝑦﷮𝑑𝑥﷯ +𝑥𝑦 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯=0 𝒙𝒚 𝒅﷮𝟐﷯𝒚﷮𝒅 𝒙﷮𝟐﷯﷯ +𝒙 𝒅𝒚﷮𝒅𝒙﷯﷯﷮𝟐﷯−𝒚 𝒅𝒚﷮𝒅𝒙﷯=𝟎

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