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Chapter 9 Class 12 Differential Equations

Serial order wise

Last updated at May 29, 2018 by Teachoo

Ex 9.5, 13 For each of the differential equations in Exercises from 11 to 15 , find the particular solution satisfying the given condition : 𝑥 sin2 𝑦𝑥−𝑦𝑑𝑥+𝑥𝑑𝑦=0;𝑦= 𝜋4 When 𝑥=1 Differential equation can be written as 𝑥 sin2 𝑦𝑥−𝑦𝑑𝑥+𝑥𝑑𝑦=0 x dy = − 𝑥 sin2 𝑦𝑥−𝑦dx 𝑑𝑦𝑑𝑥 = − 𝑥𝑥 sin2 𝑦𝑥 − 𝑦𝑥 Let F(x, y) = 𝑑𝑦𝑑𝑥 =− 𝑥𝑥 sin2 𝑦𝑥 − 𝑦𝑥 Finding F(𝜆x, 𝜆y) F(𝜆x, 𝜆y) = − sin2 𝜆𝑦𝜆𝑥− 𝜆𝑦𝜆𝑥 = − sin2 𝑦𝑥− 𝑦𝑥 = 𝜆° F (x, y) ∴ F(x, y) is 𝑎 homogenous function of degree zero Putting y = vx Diff w.r.t. x 𝑑𝑦𝑑𝑥 = x 𝑑𝑣𝑑𝑥 + v Putting value of 𝑑𝑦𝑑𝑥 and y = vx in (1) 𝑑𝑦𝑑𝑥 = − 𝑥𝑥 sin2 𝑦𝑥 − 𝑦𝑥 v + 𝑥 𝑑𝑣𝑑𝑥 = − sin2 𝑣𝑥𝑥− 𝑣𝑥𝑥 v + 𝑥 𝑑𝑣𝑑𝑥 = − sin2𝑣−𝑣 𝑥 𝑑𝑣𝑑𝑥 = v − sin2 v − v 𝑥 𝑑𝑣𝑑𝑥 = − sin2 v 𝑑𝑣 sin2𝑣 = − 𝑑𝑥𝑥 Integrating both sides 𝑑𝑣 sin2𝑣 = − 𝑑𝑥𝑥 𝑐𝑜𝑠𝑒 𝑐2 𝑣 𝑑𝑣 = − 𝑑𝑥𝑥 − cot v = − log 𝑥+𝑐 log 𝑥 − cot v = C Putting value of v = 𝑦𝑥 log 𝑥 − cot 𝑦𝑥 = C Putting x = 1 & y = 𝜋4 log 1 − cot 𝜋4 = C 0 − 1 = C C = −1 Putting value of C in (2) log 𝑥 − cot 𝑦𝑥 = −1 cot 𝑦𝑥 − log 𝑥 = 1 cot 𝑦𝑥 − log 𝑥 = log e cot 𝑦𝑥 = log 𝑥 + log e cot 𝒚𝒙 = log 𝒆𝒙