Ex 9.3, 2 - Chapter 9 Class 12 Differential Equations

Transcript

Ex 9.3, 2 Form a differential equation representing the given family of curves by eliminating arbitrary constants 𝑎 and 𝑏. 𝑦﷮2﷯=𝑎 𝑏﷮2﷯− 𝑥﷮2﷯﷯ 𝑦﷮2﷯=𝑎 𝑏﷮2﷯− 𝑥﷮2﷯﷯ 𝑦﷮2﷯=𝑎 𝑏﷮2﷯−𝑎 𝑥﷮2﷯ Here, We Eliminate Constant By Differentiating 𝑦 ∴ Diff. Both Sides w.r.t. 𝑥 2𝑦. 𝑑𝑦﷮𝑑𝑥﷯=0−2𝑎𝑥 2𝑦 𝑑𝑦﷮𝑑𝑥﷯=−2𝑎𝑥 𝑦 𝑑𝑦﷮𝑑𝑥﷯=−𝑎𝑥 − 𝑦﷮𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑎 𝑎 = −𝑦﷮𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ Now, 𝑦 𝑑𝑦﷮𝑑𝑥﷯=−𝑎𝑥 Again Differentiating w.r.t. 𝑥 𝑑𝑦﷮𝑑𝑥﷯. 𝑑𝑦﷮𝑑𝑥﷯+𝑦. 𝑑﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯=−𝑎 𝑑𝑥﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯﷮2﷯+𝑦. 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯=−𝑎 𝑑𝑦﷮𝑑𝑥﷯﷯﷮2﷯+𝑦. 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯= 𝑦﷮𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ 𝑦﷮′﷯﷯﷮2﷯+𝑦. 𝑦﷮′′﷯= 𝑦﷮𝑥﷯. 𝑦﷮′﷯ 𝑥 𝑦﷮′﷯﷯﷮2﷯+𝑥𝑦 𝑦﷮′′﷯=𝑦. 𝑦﷮′﷯ 𝒙𝒚 𝒚﷮′′﷯+𝒙 𝒚﷮′﷯﷯﷮𝟐﷯−𝒚. 𝒚﷮′﷯=𝟎 which is the required differential equation