Misc 3 - Chapter 9 Class 12 Differential Equations

Transcript

Misc 3 From the differential equation representing the family of curves given by 𝑥−𝑎﷯﷮2﷯+2 𝑦﷮2﷯= 𝑎﷮2﷯, where 𝑎 is an arbitrary constant Number of time we Differentiate is equal to number of constants. 𝑥−𝑎﷯﷮2﷯+2 𝑦﷮2﷯= 𝑎﷮2﷯ Differentiating w.r.t. 𝑥 Both Sides 𝑑 𝑥 − 𝑎﷯﷮2﷯ + 2 𝑦﷮2﷯﷯﷮𝑑𝑥﷯= 𝑑( 𝑎﷮2﷯)﷮𝑑𝑥﷯ 2 𝑥−𝑎﷯+2.2𝑦 𝑑𝑦﷮𝑑𝑥﷯=0 4𝑦 𝑑𝑦﷮𝑑𝑥﷯=−2 𝑥−𝑎﷯ 2𝑦 𝑑𝑦﷮𝑑𝑥﷯=− 𝑥−𝑎﷯ 2𝑦 𝑑𝑦﷮𝑑𝑥﷯=𝑎−𝑥 a = 2𝑦 𝑑𝑦﷮𝑑𝑥﷯+𝑥 Putting Value of in 𝑥−𝑎﷯﷮2﷯+2 𝑦﷮2﷯= 𝑎﷮2﷯ 𝑥− 2𝑦 𝑑𝑦﷮𝑑𝑥﷯+𝑥﷯﷯﷮2﷯+2 𝑦﷮2﷯= 2𝑦 𝑑𝑦﷮𝑑𝑥﷯+𝑥﷯﷮2﷯ 𝑥−𝑥−2𝑦 𝑑𝑦﷮𝑑𝑥﷯﷯﷮2﷯+2 y﷮2﷯= 2𝑦 𝑑𝑦﷮𝑑𝑥﷯+𝑥﷯﷮2﷯ −2𝑦 𝑑𝑦﷮𝑑𝑥﷯﷯﷮2﷯+2 𝑦﷮2﷯= 2𝑦 𝑑𝑦﷮𝑑𝑥﷯﷯﷮2﷯+ 𝑥﷮2﷯+2 . 2𝑦 𝑑𝑦﷮𝑑𝑥﷯﷯ . 𝑥 2𝑦 𝑑𝑦﷮𝑑𝑥﷯﷯﷮2﷯− 2𝑦 𝑑𝑦﷮𝑑𝑥﷯﷯﷮2﷯+2 𝑦﷮2﷯= 𝑥﷮2﷯+2𝑥 . 2𝑦 𝑑𝑦﷮𝑑𝑥﷯ 2 𝑦﷮2﷯− 𝑥﷮2﷯=4𝑥𝑦 𝑑𝑦﷮𝑑𝑥﷯ 2 𝑦﷮2﷯− 𝑥﷮2﷯﷮4𝑥𝑦﷯= 𝑑𝑦﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯= 2 𝑦﷮2﷯ − 𝑥﷮2﷯﷮4𝑥𝑦﷯ 𝒚﷮′﷯= 𝟐 𝒚﷮𝟐﷯ − 𝒙﷮𝟐﷯﷮𝟒𝒙𝒚﷯