# Ex 9.5, 2

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

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

Chapter 9 Class 12 Differential Equations

Serial order wise

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CA Maninder Singh

CA Maninder Singh is a Chartered Accountant for the past 8 years. He provides courses for Practical Accounts, Taxation and Efiling at teachoo.com .