Question 6 - Formation of Differntial equation when general solution given - Chapter 9 Class 12 Differential Equations
Last updated at April 16, 2024 by Teachoo
Formation of Differntial equation when general solution given
Formation of Differntial equation when general solution given
Last updated at April 16, 2024 by Teachoo
Question 6 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−𝑎2= 𝑎2 x + 𝑎2+ 𝑦−𝑎2= 𝑎2 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. 𝑑𝑦𝑑𝑥 + 2𝑎 − 2a 𝑑𝑦𝑑𝑥 + 0 = 0 x + y. 𝑑𝑦𝑑𝑥 + 𝑎 − 𝑎𝑑𝑦𝑑𝑥 = 0 x + y. 𝑑𝑦𝑑𝑥 = − 𝑎 + 𝑎𝑑𝑦𝑑𝑥 x + y 𝑑𝑦𝑑𝑥 = 𝑎 𝑑𝑦𝑑𝑥 −1 𝑎 = 𝑥 + 𝑦 𝑑𝑦𝑑𝑥 𝑑𝑦𝑑𝑥 − 1 𝑎 = 𝒙 + 𝒚 𝒚′ 𝒚′ − 𝟏 Putting value of a in (1) x−(−𝑎)2+ 𝑦−𝑎2= 𝑎2 x− − 𝑥 + 𝑦 𝑦′ 𝑦′ − 12+ 𝑦− 𝑥 + 𝑦 𝑦′ 𝑦′ − 12= 𝑥 + 𝑦 𝑦′ 𝑦′ − 12 x+ 𝑥 + 𝑦 𝑦′ 𝑦′ − 12+ 𝑦− 𝑥 + 𝑦 𝑦′ 𝑦′ − 12= 𝑥 + 𝑦 𝑦′ 𝑦′ − 12 𝑥 𝑦′− 1 + 𝑥 + 𝑦 𝑦′ 𝑦′ − 12+ 𝑦 𝑦′− 1 − 𝑥 − 𝑦 𝑦′ 𝑦′ − 12= 𝑥 + 𝑦 𝑦′ 𝑦′ − 12 𝑥 𝑦′ − 𝑥 + 𝑥 + 𝑦 𝑦′ 𝑦′ − 12+ −𝑥 − 𝑦 𝑦′ − 12= 𝑥 + 𝑦 𝑦′ 𝑦′ − 12 (𝑥 + 𝑦) 𝑦′ 𝑦′ − 12+ −(𝑥 + 𝑦) 𝑦′ − 12= 𝑥 + 𝑦 𝑦′ 𝑦′ − 12 (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