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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
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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. 𝑦﷮′﷯= 𝑥+𝑦﷮𝑥﷯ Step 1: Find 𝑑𝑦﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑥 + 𝑦﷮𝑥﷯ Step 2. Putting F(x, y) = 𝑑𝑦﷮𝑑𝑥﷯ and find F(𝜆x, 𝜆y) So, F(x, y) = 𝑥 + 𝑦﷮𝑥﷯ F(𝜆x, 𝜆y) = 𝜆𝑥 +𝜆𝑦﷮𝜆𝑥﷯ = 𝜆(𝑥 +𝑦)﷮𝜆𝑥﷯ = 𝑥 + 𝑦﷮𝑥﷯ = F(x, y) = 𝜆°F(x, y) Therefore F(x, y) Is a homogenous function of degree zero. Hence 𝑑𝑦﷮𝑑𝑥﷯ is a homogenous differential equation Step 3: Solving 𝑑𝑦﷮𝑑𝑥﷯ by putting y = vx Put y = vx. differentiating w.r.t.x 𝑑𝑦﷮𝑑𝑥﷯ = x 𝑑𝑣﷮𝑑𝑥﷯+ 𝑣𝑑𝑥﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑥 𝑑𝑣﷮𝑑𝑥﷯ + v Putting value of 𝑑𝑦﷮𝑑𝑥﷯ and y = vx in (1) 𝑑𝑦﷮𝑑𝑥﷯ = 𝑥 + 𝑦﷮𝑥﷯ 𝑥 𝑑𝑣﷮𝑑𝑥﷯ + v = 𝑥 + 𝑣𝑥﷮𝑥﷯ 𝑥 𝑑𝑣﷮𝑑𝑥﷯ + v = 1+𝑣 𝑥 𝑥 𝑑𝑣﷮𝑑𝑥﷯ = 1+𝑣−𝑣 𝑥 𝑑𝑣﷮𝑑𝑥﷯ = 1 𝑑𝑣﷮𝑑𝑥﷯ = 1﷮𝑥﷯ Integrating both sides ﷮﷮𝑑𝑣= ﷮﷮ 𝑑𝑥﷮𝑥﷯ ﷯ ﷯ v = log 𝑥﷯+𝑐 Putting v = 𝑦﷮𝑥﷯ 𝒚﷮𝒙﷯ = x log 𝒙﷯ + cx is the general solution of the given differential equation

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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