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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


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Ex 9.4, 14 For each of the differential equations in Exercises from 11 to 15 , find the particular solution satisfying the given condition : 𝑑𝑦/𝑑𝑥−𝑦/𝑥+𝑐𝑜𝑠𝑒𝑐(𝑦/𝑥)=0;𝑦=0 When 𝑥=1 Differential equation is 𝑑𝑦/𝑑𝑥 = 𝑦/𝑥−𝑐𝑜𝑠𝑒𝑐(𝑦/𝑥) Let F(x, y) = 𝑑𝑦/𝑑𝑥 = 𝑦/𝑥−𝑐𝑜𝑠𝑒𝑐(𝑦/𝑥) Finding F(𝝀x, 𝝀y) F(𝜆x, 𝜆y) = ("𝜆" 𝑦)/("𝜆" 𝑥)−𝑐𝑜𝑠𝑒𝑐(("𝜆" 𝑦)/("𝜆" 𝑥)) = 𝑦/𝑥 − cosec (𝑦/𝑥) = 𝜆° F(x, y) ∴ F(x, y) is 𝑎 homogenous function of degree zero F(𝜆x, 𝜆y) = 𝜆° F(x , y) Putting y = vx Diff w.r.t. x 𝒅𝒚/𝒅𝒙 = x 𝒅𝒗/𝒅𝒙 + v Putting value of 𝑑𝑦/𝑑𝑥 and y = vx in (1) 𝑑𝑦/𝑑𝑥 = 𝑦/𝑥−𝑐𝑜𝑠𝑒𝑐(𝑦/𝑥) v + x 𝒅𝒗/𝒅𝒙 = 𝒗𝒙/𝒙 − cosec (𝒗𝒙/𝒙) v + x 𝑑𝑣/𝑑𝑥 = v − cosec v (𝑥 𝑑𝑣)/𝑑𝑥 = v − cosec v − v (𝑥 𝑑𝑣)/𝑑𝑥 = − cosec v (−𝒅𝒗)/(𝒄𝒐𝒔𝒆𝒄 𝒗) = 𝒅𝒙/𝒙 Integrating both sides ∫1▒〖(−𝑑𝑣)/(𝑐𝑜𝑠𝑒𝑐 𝑣) " = " ∫1▒𝑑𝑥/𝑥〗 ∫1▒〖−sin⁡𝑣 𝑑𝑣〗=log⁡〖|𝑥|+𝑐〗 Put value of v = 𝑦/𝑥 cos 𝒚/𝒙 = log |𝒙| + C Putting x = 1 & y = 0 cos 0/1 = log 1 + C 1 = 0 + C C = 1 Putting value in (2) cos 𝑦/2 = log |𝑥| + 1 cos 𝑦/𝑥 = log |𝑥| + log e cos 𝒚/𝒙 = log |𝒆𝒙|

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.