Ex 9.5, 8 - Show homogeneous: x dy/dx - y + x sin (y/x) = 0 - Ex 9.5

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Ex 9.5, 8 show that the given differential equation is homogeneous and solve each of them. 𝑥 𝑑𝑦﷮𝑑𝑥﷯−𝑦+𝑥𝑠𝑖𝑛 𝑦﷮𝑥﷯﷯=0 Step 1: Find 𝑑𝑦﷮𝑑𝑥﷯ 𝑥 𝑑𝑦﷮𝑑𝑥﷯ = y − x sin 𝑦﷮𝑥﷯﷯ Step 2. Put 𝑑𝑦﷮𝑑𝑥﷯ = F (x, y) and find F(𝜆x, 𝜆y) F(x, y) = 𝑦﷮𝑥﷯ − sin 𝑦﷮𝑥﷯﷯ F(𝜆x, 𝜆y) = 𝜆𝑦﷮𝜆𝑥﷯ − sin 𝜆𝑦﷮𝜆𝑥﷯﷯ = 𝑦﷮𝑥﷯ − sin 𝑦﷮𝑥﷯﷯ = F(x, y) = 𝜆° 𝐹(𝑥, 𝑦)﷯ ∴ F (x, y) is a homogenous function of degree 0 . So the differential equation 𝑑𝑦﷮𝑑𝑥﷯ is homogenous Step 3 : Let y = vx Solving 𝑑𝑦﷮𝑑𝑥﷯= 𝑦﷮𝑥﷯ - sin 𝑦﷮𝑥﷯﷯ Putting y = vx Diff w.r.t.x 𝑑𝑦﷮𝑑𝑥﷯ = x 𝑑𝑣﷮𝑑𝑥﷯ + v 𝑑𝑥﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = x 𝑑𝑣﷮𝑑𝑥﷯ + v Putting value of 𝑑𝑦﷮𝑑𝑥﷯ = 𝑥2 + 𝑦﷮2﷯﷮𝑥2 + 𝑥𝑦﷯ and y = vx in (1) 𝑥 𝑑𝑦﷮𝑑𝑥﷯ = y − x sin 𝑦﷮𝑥﷯﷯ v + 𝑥 𝑑𝑣﷮𝑑𝑥﷯ = 𝑣𝑥﷮𝑥﷯ − sin 𝑣𝑥﷮𝑥﷯﷯ v + 𝑥 𝑑𝑣﷮𝑑𝑥﷯ = 𝑣 − sin v 𝑥 𝑑𝑣﷮𝑑𝑥﷯ = v − sin v − v 𝑥 𝑑𝑣﷮𝑑𝑥﷯ = − sin﷮𝑣﷯ 𝑑𝑣﷮𝑑𝑥﷯ = − sin﷮𝑣﷯﷮𝑥﷯ 𝑑𝑣﷮𝑠𝑖𝑛 𝑣﷯ = −𝑑𝑥﷮𝑥﷯ Integrating both sides ﷮﷮ 𝑑𝑣﷮𝑠𝑖𝑛 𝑣﷯= ﷮﷮ −𝑑𝑥﷮𝑥﷯﷯﷯ ﷮﷮𝑐𝑜𝑠𝑒𝑐 𝑣 𝑑𝑣=− ﷮﷮ 𝑑𝑥﷮𝑥﷯﷯ ﷯ log 𝑐𝑜𝑠𝑒𝑐 𝑣 − cot﷮𝑥﷯﷯=− log﷮ 𝑥﷯﷯+ log﷮𝑐﷯ log 𝑐𝑜𝑠𝑒𝑐 𝑣 − cot﷮𝑣﷯﷯+ log﷮ 𝑥﷯﷯= log﷮𝑐﷯ log 𝑥(𝑐𝑜𝑠𝑒𝑐 𝑣 − cot﷮𝑣)﷯﷯= log﷮𝑐﷯ x (cosec v − cot v) = C x 1﷮ sin﷮𝑣﷯﷯− cos﷮𝑣﷯﷮ sin﷮𝑣﷯﷯﷯ = C x (1− cos﷮𝑣)﷯﷮ sin﷮𝑣﷯﷯ = C x(1 − cos v) = C sin v Putting value of v = 𝑦﷮𝑥﷯ x 𝟏−𝒄𝒐𝒔 𝒚﷮𝒙﷯﷯﷯ = C sin 𝒚﷮𝒙﷯﷯

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