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


Transcript

Ex 9.4, 1 In each of the Exercise 1 to 10 , show that the given differential equation is homogeneous and solve (x^2+xy)dy=(x^2+y^2 )𝑑x (𝑥^2+𝑥𝑦)𝑑𝑦 =(𝑥^2+𝑦^2 )𝑑𝑥 Step 1: Find 𝑑𝑦/𝑑𝑥 (x2 + xy)dy = (𝑥^2+𝑦^2 )𝑑𝑥 𝒅𝒚/𝒅𝒙 = (𝒙^𝟐 + 𝒚^𝟐)/(𝒙^𝟐 + 𝒙𝒚) Step 2: Putting F(x , y) = 𝑑𝑦/𝑑𝑥 and finding F(𝜆x, 𝜆y) F(x, y) = (𝑥^2 + 𝑦^2)/(𝑥^2 + 𝑥𝑦) Finding F(𝝀x, 𝝀y) F(𝜆x, 𝜆y) = ((𝜆〖𝑥)〗^2+(𝜆〖𝑦)〗^2)/((𝜆〖𝑥)〗^2+𝜆𝑥 + 𝜆𝑦) = (𝜆^2 𝑥^2 + 𝜆^2 𝑦^2)/(𝜆^2 𝑥^2 + 𝜆^2 𝑥𝑦) = (𝜆^2 (𝑥^2 𝑦^2))/(𝜆^2 (𝑥^2+ 𝑥𝑦)) = (𝑥^2 𝑦^2)/(𝑥^2 + 𝑥𝑦) = F(x, y) So, F(𝜆x, 𝜆y) = F(x, y) = 𝜆0 F (x, y) Thus, F(x, y) is a homogenous equation function of order zero Therefore 𝑑𝑦/𝑑𝑥 is a homogenous differential equation Step 3 : Solving 𝑑𝑦/𝑑𝑥 by putting y = vx Putting y = vx Diff w.r.t.x 𝑑𝑦/𝑑𝑥 = x𝑑𝑣/𝑑𝑥 + v 𝑑𝑥/𝑑𝑥 𝒅𝒚/𝒅𝒙 = x 𝒅𝒗/𝒅𝒙 + v Putting value of 𝑑𝑦/𝑑𝑥 and y = vx in (1) 𝑑𝑦/𝑑𝑥 = (𝑥^2 + 𝑦^2)/(𝑥^2 + 𝑥𝑦) x 𝒅𝒗/𝒅𝒙 + v = (𝒙𝟐 + (𝒗𝒙)^𝟐)/(𝒙𝟐 + 𝒙(𝒗𝒙)) x 𝑑𝑣/𝑑𝑥 + v = (𝒙𝟐(𝟏 + 𝒗^𝟐))/(𝑥2 + 𝑥2𝑣) x 𝑑𝑣/𝑑𝑥 + v = (𝒙𝟐(1 + 𝑣^2))/(𝒙𝟐(1 + 𝑣)) x 𝑑𝑣/𝑑𝑥 + v = (1 + 𝑣^2)/(1 + 𝑣) x 𝑑𝑣/𝑑𝑥 = (1 + 𝑣^2)/(1 + 𝑣)−𝒗 x 𝑑𝑣/𝑑𝑥 = (1 + 𝑣^2 − 𝑣 − 𝑣^2)/(1+𝑣) x 𝒅𝒗/𝒅𝒙 = (𝟏 − 𝒗)/(𝟏 + 𝒗) ((1 + 𝑣))/((1 − 𝑣)) dv = 𝑑𝑥/𝑥 −((𝑣 + 1)/(𝑣 − 1)) dv = 𝑑𝑥/𝑥 ((𝒗 + 𝟏)/(𝒗 − 𝟏)) dv = (−𝒅𝒙)/𝒙 Integrating both sides ∫1▒((𝑣 + 1)/(𝑣 − 1)) 𝑑𝑣=−∫1▒𝑑𝑥/𝑥 ∫1▒((𝒗 + 𝟏)/(𝒗 − 𝟏)) 𝒅𝒗 = −log|𝒙|+𝒄 Let I = ∫1▒((𝑣 + 1)/(𝑣 − 1)) 𝑑𝑣 Solving I I = ∫1▒((𝑣 + 1 − 1 + 1)/(𝑣 − 1)) 𝑑𝑣 I = ∫1▒((𝒗 − 𝟏 + 𝟐)/(𝒗 − 𝟏)) 𝑑𝑣 I = ∫1▒((𝒗 − 𝟏 )/(𝒗 − 𝟏)+𝟐/(𝒗 − 𝟏)) 𝑑𝑣 I = ∫1▒(1+2/(𝑣 − 1)) 𝑑𝑣 I = ∫1▒𝑑𝑣+∫1▒2/(𝑣 − 1) 𝑑𝑣 I = 𝒗+𝟐 𝐥𝐨𝐠⁡〖|𝒗−𝟏|〗 Putting v = y/x I = 𝑦/𝑥+2 log⁡|𝑦/𝑥−1| I = 𝒚/𝒙+𝟐 𝒍𝒐𝒈⁡|(𝒚 − 𝒙)/𝒙| Putting value of I in (2) 𝒚/𝒙+𝟐 𝒍𝒐𝒈⁡|(𝒚 − 𝒙)/𝒙|=−𝒍𝒐𝒈⁡|𝒙|+𝑪 𝑦/𝑥+𝒍𝒐𝒈⁡〖|(𝒚 − 𝒙)/𝒙|^𝟐 〗+log⁡|𝑥|=𝐶 𝑦/𝑥+log⁡|(𝑦 − 𝑥)^2/𝑥^2 |+log⁡|𝑥|=𝐶 𝑦/𝑥+log⁡|(𝒚 − 𝒙)^𝟐/𝒙^𝟐 × 𝒙|=𝐶 𝑦/𝑥+log⁡|(𝑦 − 𝑥)^2/𝑥|=𝐶 𝒍𝒐𝒈⁡|(𝒚 − 𝒙)^𝟐/𝒙|=𝐶−𝑦/𝑥 (𝒚 − 𝒙)^𝟐/𝒙= 𝒆^(𝑪 − 𝒚/𝒙) (𝑦 − 𝑥)^2/𝑥 = 𝑒^𝑐 × 𝑒^(− 𝑦/𝑥) (𝑦 − 𝑥)^2/𝑥 = 𝑐 𝑒^(− 𝑦/𝑥) (𝒙−𝒚)^𝟐 = 𝒄𝒙 𝒆^(− 𝒚/𝒙) (𝐴𝑠 𝑎 log⁡𝑏=log⁡〖𝑏^𝑎 〗) (𝐴𝑠 log⁡𝑎+log⁡𝑏=log⁡𝑎𝑏)

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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.