    1. Chapter 9 Class 12 Differential Equations (Term 2)
2. Serial order wise
3. Examples

Transcript

Example 25 From the differential equation of the family of circles in the second quadrant and touching the coordinate axes . Drawing figure : Let C be the family of circles in second quadrant touching coordinate. Let radius be 𝑎 ∴ Center of circle = (−𝑎, 𝑎) Equation representing family C is x−(−𝑎)﷯﷮2﷯+ 𝑦−𝑎﷯﷮2﷯= 𝑎﷮2﷯ x + 𝑎﷯﷮2﷯+ 𝑦−𝑎﷯﷮2﷯= 𝑎﷮2﷯ 𝑥2 + 𝑎2 + 2ax + y2 + 𝑎2 − 2𝑎y = 𝑎2 𝑥2 + 𝑦2 + 2ax − 2ay + 2𝑎2 = 𝑎2 𝑥2 + y2 + 2𝑎x − 2𝑎y + 𝑎2 = 0 Differentiate w.r.t x 2x + 2y. 𝑑𝑦﷮𝑑𝑥﷯ + 2𝑎 − 2a 𝑑𝑦﷮𝑑𝑥﷯ + 0 = 0 x + y. 𝑑𝑦﷮𝑑𝑥﷯ + 𝑎 − 𝑎𝑑𝑦﷮𝑑𝑥﷯ = 0 x + y. 𝑑𝑦﷮𝑑𝑥﷯ = − 𝑎 + 𝑎𝑑𝑦﷮𝑑𝑥﷯ x + y 𝑑𝑦﷮𝑑𝑥﷯ = 𝑎 𝑑𝑦﷮𝑑𝑥﷯ −1﷯ 𝑎 = 𝑥 + 𝑦 𝑑𝑦﷮𝑑𝑥﷯ ﷮ 𝑑𝑦﷮𝑑𝑥﷯ − 1﷯ 𝑎 = 𝒙 + 𝒚 𝒚﷮′﷯ ﷮ 𝒚﷮′﷯ − 𝟏﷯ Putting value of a in (1) x−(−𝑎)﷯﷮2﷯+ 𝑦−𝑎﷯﷮2﷯= 𝑎﷮2﷯ x− − 𝑥 + 𝑦 𝑦﷮′﷯ ﷮ 𝑦﷮′﷯ − 1﷯﷯﷯﷮2﷯+ 𝑦− 𝑥 + 𝑦 𝑦﷮′﷯ ﷮ 𝑦﷮′﷯ − 1﷯﷯﷮2﷯= 𝑥 + 𝑦 𝑦﷮′﷯ ﷮ 𝑦﷮′﷯ − 1﷯﷯﷮2﷯ x+ 𝑥 + 𝑦 𝑦﷮′﷯ ﷮ 𝑦﷮′﷯ − 1﷯﷯﷮2﷯+ 𝑦− 𝑥 + 𝑦 𝑦﷮′﷯ ﷮ 𝑦﷮′﷯ − 1﷯﷯﷮2﷯= 𝑥 + 𝑦 𝑦﷮′﷯ ﷮ 𝑦﷮′﷯ − 1﷯﷯﷮2﷯ 𝑥 𝑦﷮′﷯− 1﷯ + 𝑥 + 𝑦 𝑦﷮′﷯ ﷮ 𝑦﷮′﷯ − 1﷯﷯﷮2﷯+ 𝑦 𝑦﷮′﷯− 1﷯ − 𝑥 − 𝑦 𝑦﷮′﷯ ﷮ 𝑦﷮′﷯ − 1﷯﷯﷮2﷯= 𝑥 + 𝑦 𝑦﷮′﷯ ﷮ 𝑦﷮′﷯ − 1﷯﷯﷮2﷯ 𝑥 𝑦﷮′﷯ − 𝑥 + 𝑥 + 𝑦 𝑦﷮′﷯ ﷮ 𝑦﷮′﷯ − 1﷯﷯﷮2﷯+ −𝑥 − 𝑦 ﷮ 𝑦﷮′﷯ − 1﷯﷯﷮2﷯= 𝑥 + 𝑦 𝑦﷮′﷯ ﷮ 𝑦﷮′﷯ − 1﷯﷯﷮2﷯ (𝑥 + 𝑦) 𝑦﷮′﷯ ﷮ 𝑦﷮′﷯ − 1﷯﷯﷮2﷯+ −(𝑥 + 𝑦) ﷮ 𝑦﷮′﷯ − 1﷯﷯﷮2﷯= 𝑥 + 𝑦 𝑦﷮′﷯ ﷮ 𝑦﷮′﷯ − 1﷯﷯﷮2﷯ (𝑥 + 𝑦) ﷮2﷯ ( 𝑦﷮′﷯)﷮2﷯+ (𝑥 + 𝑦) ﷮2﷯= 𝑥 + 𝑦 𝑦﷮′﷯﷯﷮2﷯ (𝒙 + 𝒚) ﷮𝟐﷯ ( 𝒚﷮′﷯)﷮𝟐﷯ + 𝟏﷯= 𝒙 + 𝒚 𝒚﷮′﷯﷯﷮𝟐﷯ which is the required differential equation 