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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Ex 9.4, 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) = ("𝜆" 𝑦)/("𝜆" 𝑥) − sin (("𝜆" 𝑦)/("𝜆" 𝑥)) = 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 ∫1▒〖𝑑𝑣/(𝑠𝑖𝑛 𝑣)=∫1▒(−𝑑𝑥)/𝑥〗 ∫1▒〖𝑐𝑜𝑠𝑒𝑐 𝑣 𝑑𝑣=−∫1▒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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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.