Ex 9.4, 2 - Chapter 9 Class 12 Differential Equations
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Ex 9.4, 2 In each of the Exercise 1 to 10 , show that the given differential equation is homogeneous and solve each of them. ๐ฆ^โฒ=(๐ฅ+๐ฆ)/๐ฅ Step 1: Find ๐๐ฆ/๐๐ฅ ๐๐ฆ/๐๐ฅ = (๐ฅ + ๐ฆ)/๐ฅ Step 2: Putting F(x, y) = ๐๐ฆ/๐๐ฅ and find F(๐x, ๐y) So, F(x, y) = (๐ + ๐)/๐ F(๐x, ๐y) = (๐๐ฅ +๐๐ฆ)/๐๐ฅ = (๐(๐ฅ +๐ฆ))/๐๐ฅ = (๐ฅ + ๐ฆ)/๐ฅ = F(x, y) = ๐ยฐF(x, y) Therefore F(x, y) is a homogenous function of degree zero. Hence ๐๐ฆ/๐๐ฅ is a homogenous differential equation Step 3: Solving ๐๐ฆ/๐๐ฅ by putting y = vx Put y = vx. differentiating w.r.t.x ๐๐ฆ/๐๐ฅ = x ๐๐ฃ/๐๐ฅ+๐ฃ๐๐ฅ/๐๐ฅ ๐ ๐/๐ ๐ = ๐ ๐ ๐/๐ ๐ + v Putting value of ๐๐ฆ/๐๐ฅ and y = vx in (1) ๐๐ฆ/๐๐ฅ = (๐ฅ + ๐ฆ)/๐ฅ ๐ ( ๐ ๐)/๐ ๐ + v = (๐ + ๐๐)/๐ ๐ฅ ( ๐๐ฃ)/๐๐ฅ + v = 1+๐ฃ ๐ฅ (๐ฅ ๐๐ฃ)/๐๐ฅ = 1+๐ฃโ๐ฃ ๐ฅ ( ๐๐ฃ)/๐๐ฅ = 1 ( ๐ ๐)/๐ ๐ = ๐/๐ Integrating both sides โซ1โใ๐๐ฃ=โซ1โใ๐๐ฅ/๐ฅ ใ ใ v = log|๐|+๐ Putting v = ๐ฆ/๐ฅ ๐ฆ/๐ฅ = log|๐ฅ| + c y = x log|๐| + cx
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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo