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
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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. ๐‘ฆ๏ทฎโ€ฒ๏ทฏ= ๐‘ฅ+๐‘ฆ๏ทฎ๐‘ฅ๏ทฏ Step 1: Find ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘ฅ + ๐‘ฆ๏ทฎ๐‘ฅ๏ทฏ Step 2. Putting F(x, y) = ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ and find F(๐œ†x, ๐œ†y) So, F(x, y) = ๐‘ฅ + ๐‘ฆ๏ทฎ๐‘ฅ๏ทฏ F(๐œ†x, ๐œ†y) = ๐œ†๐‘ฅ +๐œ†๐‘ฆ๏ทฎ๐œ†๐‘ฅ๏ทฏ = ๐œ†(๐‘ฅ +๐‘ฆ)๏ทฎ๐œ†๐‘ฅ๏ทฏ = ๐‘ฅ + ๐‘ฆ๏ทฎ๐‘ฅ๏ทฏ = F(x, y) = ๐œ†ยฐF(x, y) Therefore F(x, y) Is a homogenous function of degree zero. Hence ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ is a homogenous differential equation Step 3: Solving ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ by putting y = vx Put y = vx. differentiating w.r.t.x ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = x ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ+ ๐‘ฃ๐‘‘๐‘ฅ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘ฅ ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + v Putting value of ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ and y = vx in (1) ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘ฅ + ๐‘ฆ๏ทฎ๐‘ฅ๏ทฏ ๐‘ฅ ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + v = ๐‘ฅ + ๐‘ฃ๐‘ฅ๏ทฎ๐‘ฅ๏ทฏ ๐‘ฅ ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + v = 1+๐‘ฃ ๐‘ฅ ๐‘ฅ ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = 1+๐‘ฃโˆ’๐‘ฃ ๐‘ฅ ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = 1 ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = 1๏ทฎ๐‘ฅ๏ทฏ Integrating both sides ๏ทฎ๏ทฎ๐‘‘๐‘ฃ= ๏ทฎ๏ทฎ ๐‘‘๐‘ฅ๏ทฎ๐‘ฅ๏ทฏ ๏ทฏ ๏ทฏ v = log ๐‘ฅ๏ทฏ+๐‘ Putting v = ๐‘ฆ๏ทฎ๐‘ฅ๏ทฏ ๐’š๏ทฎ๐’™๏ทฏ = x log ๐’™๏ทฏ + cx is the general solution of the given differential equation

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