Example 7 - Chapter 9 Class 12 Differential Equations

Transcript

Example 7 Form the differential equation of the family of circles touching the 𝑥−𝑎𝑥𝑖𝑠 at origin. In Our Case , Center will be on 𝑦−𝑎𝑥𝑖𝑠 So, Center 0 , 𝑏﷯ i.e. 𝑎=0 Also 𝑟𝑎𝑑𝑖𝑢𝑠=𝑏 So, Equation of Circle 𝑥−0﷯﷮2﷯+ 𝑦−𝑏﷯﷮2﷯= 𝑏﷮2﷯ 𝑥﷮2﷯+ 𝑦−𝑏﷯﷮2﷯= 𝑏﷮2﷯ 𝑥﷮2﷯+ 𝑦﷮2﷯−2𝑦𝑏+ 𝑏﷮2﷯= 𝑏﷮2﷯ 𝑥﷮2﷯+ 𝑦﷮2﷯−2𝑦𝑏=0 Since there is one variable b, we differentiate once Diff. w.r.t. 𝑥 2𝑥+2𝑦 𝑑𝑦﷮𝑑𝑥﷯=2𝑏. 𝑑𝑦﷮𝑑𝑥﷯ 𝑥+𝑦 𝑑𝑦﷮𝑑𝑥﷯=𝑏. 𝑑𝑦﷮𝑑𝑥﷯ 𝑥 + 𝑦 𝑑𝑦﷮𝑑𝑥﷯﷮ 𝑑𝑦﷮𝑑𝑥﷯﷯﷯=𝑏 Putting Value of 𝑏 in (1) 𝑥﷮2﷯+ 𝑦﷮2﷯=2𝑦 𝑥 + 𝑦 𝑑𝑦﷮𝑑𝑥﷯﷮ 𝑑𝑦﷮𝑑𝑥﷯﷯﷯ 𝑥﷮2﷯+ 𝑦﷮2﷯=2𝑦 𝑥 + 𝑦 𝑑𝑦﷮𝑑𝑥﷯﷮ 𝑑𝑦﷮𝑑𝑥﷯﷯﷯ 𝑥﷮2﷯+ 𝑦﷮2﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯=2𝑦 𝑥+𝑦 𝑑𝑦﷮𝑑𝑥﷯﷯ 𝑥﷮2﷯+ 𝑦﷮2﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯=2𝑦 𝑥+2 𝑦﷮2﷯ 𝑑𝑦﷮𝑑𝑥﷯ 𝑥﷮2﷯ 𝑑𝑦﷮𝑑𝑥﷯+ 𝑦﷮2﷯ 𝑑𝑦﷮𝑑𝑥﷯=2𝑥𝑦+2 𝑦﷮2﷯ 𝑑𝑦﷮𝑑𝑥﷯ 𝑥﷮2﷯ 𝑑𝑦﷮𝑑𝑥﷯+ 𝑦﷮2﷯ 𝑑𝑦﷮𝑑𝑥﷯−2 𝑦﷮2 ﷯ 𝑑𝑦﷮𝑑𝑥﷯=2𝑥𝑦 𝑥﷮2﷯ 𝑑𝑦﷮𝑑𝑥﷯− 𝑦﷮2﷯ 𝑑𝑦﷮𝑑𝑥﷯=2xy 𝑑𝑦﷮𝑑𝑥﷯ 𝑥﷮2﷯− 𝑦﷮2﷯﷯=2𝑥𝑦 𝒅𝒚﷮𝒅𝒙﷯= 𝟐𝒙𝒚﷮ 𝒙﷮𝟐﷯ − 𝒚﷮𝟐﷯﷯