Ex 9.5, 2 - Show homogeneous: y' = x+y / x - Solving homogeneous differential equation

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise

Transcript

Ex 9.5, 2 In each of the Exercise 1 to 10 , show that the given differential equation is homogeneous and solve each of them. ๐‘ฆ^โ€ฒ=(๐‘ฅ+๐‘ฆ)/๐‘ฅ Step 1: Find ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘ฅ + ๐‘ฆ)/๐‘ฅ Step 2. Putting F(x, y) = ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ and find F(๐œ†x, ๐œ†y) So, F(x, y) = (๐‘ฅ + ๐‘ฆ)/๐‘ฅ F(๐œ†x, ๐œ†y) = (๐œ†๐‘ฅ +๐œ†๐‘ฆ)/๐œ†๐‘ฅ = (๐œ†(๐‘ฅ +๐‘ฆ))/๐œ†๐‘ฅ = (๐‘ฅ + ๐‘ฆ)/๐‘ฅ = F(x, y) = ๐œ†ยฐF(x, y) Therefore F(x, y) Is a homogenous function of degree zero. Hence ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ is a homogenous differential equation Step 3: Solving ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ by putting y = vx Put y = vx. differentiating w.r.t.x ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = x ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ+๐‘ฃ๐‘‘๐‘ฅ/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘ฅ ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ + v Putting value of ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ and y = vx in (1) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘ฅ + ๐‘ฆ)/๐‘ฅ ๐‘ฅ ( ๐‘‘๐‘ฃ)/๐‘‘๐‘ฅ + v = (๐‘ฅ + ๐‘ฃ๐‘ฅ)/๐‘ฅ ๐‘ฅ ( ๐‘‘๐‘ฃ)/๐‘‘๐‘ฅ + v = 1+๐‘ฃ ๐‘ฅ (๐‘ฅ ๐‘‘๐‘ฃ)/๐‘‘๐‘ฅ = 1+๐‘ฃโˆ’๐‘ฃ ๐‘ฅ ( ๐‘‘๐‘ฃ)/๐‘‘๐‘ฅ = 1 ( ๐‘‘๐‘ฃ)/๐‘‘๐‘ฅ = 1/๐‘ฅ Integrating both sides โˆซ1โ–’ใ€–๐‘‘๐‘ฃ=โˆซ1โ–’ใ€–๐‘‘๐‘ฅ/๐‘ฅ ใ€— ใ€— v = log|๐‘ฅ|+๐‘ Putting v = ๐‘ฆ/๐‘ฅ ๐‘ฆ/๐‘ฅ = log|๐‘ฅ| + c y = x log|๐’™| + cx

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.