# Ex 9.5, 8

Last updated at March 11, 2017 by Teachoo

Last updated at March 11, 2017 by Teachoo

Transcript

Ex 9.5, 8 show that the given differential equation is homogeneous and solve each of them. 𝑥 𝑑𝑦𝑑𝑥−𝑦+𝑥𝑠𝑖𝑛 𝑦𝑥=0 Step 1: Find 𝑑𝑦𝑑𝑥 𝑥 𝑑𝑦𝑑𝑥 = y − x sin 𝑦𝑥 Step 2. Put 𝑑𝑦𝑑𝑥 = F (x, y) and find F(𝜆x, 𝜆y) F(x, y) = 𝑦𝑥 − sin 𝑦𝑥 F(𝜆x, 𝜆y) = 𝜆𝑦𝜆𝑥 − sin 𝜆𝑦𝜆𝑥 = 𝑦𝑥 − sin 𝑦𝑥 = F(x, y) = 𝜆° 𝐹(𝑥, 𝑦) ∴ F (x, y) is a homogenous function of degree 0 . So the differential equation 𝑑𝑦𝑑𝑥 is homogenous Step 3 : Let y = vx Solving 𝑑𝑦𝑑𝑥= 𝑦𝑥 - sin 𝑦𝑥 Putting y = vx Diff w.r.t.x 𝑑𝑦𝑑𝑥 = x 𝑑𝑣𝑑𝑥 + v 𝑑𝑥𝑑𝑥 𝑑𝑦𝑑𝑥 = x 𝑑𝑣𝑑𝑥 + v Putting value of 𝑑𝑦𝑑𝑥 = 𝑥2 + 𝑦2𝑥2 + 𝑥𝑦 and y = vx in (1) 𝑥 𝑑𝑦𝑑𝑥 = y − x sin 𝑦𝑥 v + 𝑥 𝑑𝑣𝑑𝑥 = 𝑣𝑥𝑥 − sin 𝑣𝑥𝑥 v + 𝑥 𝑑𝑣𝑑𝑥 = 𝑣 − sin v 𝑥 𝑑𝑣𝑑𝑥 = v − sin v − v 𝑥 𝑑𝑣𝑑𝑥 = − sin𝑣 𝑑𝑣𝑑𝑥 = − sin𝑣𝑥 𝑑𝑣𝑠𝑖𝑛 𝑣 = −𝑑𝑥𝑥 Integrating both sides 𝑑𝑣𝑠𝑖𝑛 𝑣= −𝑑𝑥𝑥 𝑐𝑜𝑠𝑒𝑐 𝑣 𝑑𝑣=− 𝑑𝑥𝑥 log 𝑐𝑜𝑠𝑒𝑐 𝑣 − cot𝑥=− log 𝑥+ log𝑐 log 𝑐𝑜𝑠𝑒𝑐 𝑣 − cot𝑣+ log 𝑥= log𝑐 log 𝑥(𝑐𝑜𝑠𝑒𝑐 𝑣 − cot𝑣)= log𝑐 x (cosec v − cot v) = C x 1 sin𝑣− cos𝑣 sin𝑣 = C x (1− cos𝑣) sin𝑣 = C x(1 − cos v) = C sin v Putting value of v = 𝑦𝑥 x 𝟏−𝒄𝒐𝒔 𝒚𝒙 = C sin 𝒚𝒙

Chapter 9 Class 12 Differential Equations

Serial order wise

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