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


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Ex 9.4, 3 In each of the Exercise 1 to 10, show that the given differential equation is homogeneous and solve each of them. (x−y)𝑑y−(x+y)𝑑𝑥=0 Step 1: Find 𝑑𝑦/𝑑𝑥 (x − y) dy − (x + y) dx = 0 (x − y) dy = (x + y) dx 𝒅𝒚/𝒅𝒙 = (𝒙 + 𝒚)/(𝒙 − 𝒚) Step 2: Put 𝑑𝑦/𝑑𝑥 = F(x, y) and find out F(𝜆x, 𝜆y) F(x, y) = (𝑥 + 𝑦)/(𝑥 − 𝑦) F(𝜆x, 𝜆y) = (𝜆𝑥 + 𝜆𝑦)/(𝜆𝑥−𝜆𝑦) = (𝜆(𝑥 + 𝑦))/(𝜆 (𝑥 − 𝑦)) = (𝑥 + 𝑦)/(𝑥 − 𝑦) = F(x, y) ∴ F(𝜆x, 𝜆y) = 𝜆0 F(x, y) Hence, F(x, y) is a homogenous Function of with degree zero So, 𝑑𝑦/𝑑𝑥 is a homogenous differential equation. Step 3: Solving 𝑑𝑦/𝑑𝑥 by putting y = vx Putting y = vx. Differentiating w.r.t. x 𝑑𝑦/𝑑𝑥 = x 𝑑𝑣/𝑑𝑥+𝑣𝑑𝑥/𝑑𝑥 𝒅𝒚/𝒅𝒙 = 𝒙 𝒅𝒗/𝒅𝒙 + v Putting value of 𝑑𝑦/𝑑𝑥 and y = vx in (1) 𝑑𝑦/𝑑𝑥 = (𝑥 + 𝑦)/(𝑥 − 𝑦) x 𝒅𝒗/𝒅𝒙 + v = (𝒙 + 𝒙𝒗)/(𝒙 − 𝒙𝒗) x 𝑑𝑣/𝑑𝑥 + v = (𝑥 (1 + 𝑣))/(𝑥(1 − 𝑣)) x 𝑑𝑣/𝑑𝑥 + v = ((1 + 𝑣))/((1 − 𝑣)) x 𝑑𝑣/𝑑𝑥 = ((1 + 𝑣))/((1 − 𝑣)) − v x 𝑑𝑣/𝑑𝑥 = (1 + 𝑣 − 𝑣(1 − 𝑣))/(1 − 𝑣) x 𝑑𝑣/𝑑𝑥 = (1 + 𝑣 − 𝑣 + 𝑣^2)/(1 − 𝑣) x 𝑑𝑣/𝑑𝑥 = (1 + 𝑣^2)/(1 − 𝑣) (𝟏 − 𝒗)𝒅𝒗/(𝟏 + 𝒗^𝟐 ) = 𝒅𝒙/𝒙 Integrating both sides ∫1▒((1 − 𝑣)/(1 + 𝑣^2 )) 𝑑𝑣=∫1▒𝑑𝑥/𝑥 ∫1▒𝑑𝑣/(1 + 𝑣^2 )−∫1▒(𝑣 𝑑𝑣)/(1 + 𝑣^2 )=∫1▒𝑑𝑥/𝑥 tan−1 v − ∫1▒𝒗/(𝟏 + 𝒗^𝟐 ) = log|𝒙|+𝒄 Let I = ∫1▒𝒗/(𝟏 + 𝒗^𝟐 ) dv Putting t = 1 + 𝒗^𝟐 Diff w.r.t. v 𝑑/𝑑𝑣(1 + v2) = 𝑑𝑡/𝑑𝑣 2v = 𝑑𝑡/𝑑𝑣 dv = 𝒅𝒕/𝟐𝒗 Therefore I = ∫1▒𝑣/(1 + 𝑣^2 ) dv = ∫1▒𝑑𝑡/2𝑡 = 1/2 𝑙𝑜𝑔|𝑡|+𝑐 Putting back t = 1 + v2 = 1/2 𝑙𝑜𝑔|1+𝑣^2 | + c (As ∫1▒1/(1 + 𝑥^2 ) dx = tan−1 x) Putting value of I in (2) tan−1 v − ∫1▒𝒗/(𝟏 + 𝒗^𝟐 ) = log|𝑥|+𝑐 tan−1 v "− " 𝟏/𝟐 log |𝟏+𝒗𝟐| = log |𝑥| + c tan−1 v "= " 1/2 log |1+𝑣2| + log |𝑥| + c tan−1 v "= " 1/2 log |1+𝑣2| + 2/2 log |𝑥| + c tan−1 v "= " 1/2 ["log " |1+𝑣2|" + " 2" log " |𝑥|] + c tan−1 v "= " 𝟏/𝟐 "log" [" " |𝟏+𝒗𝟐|.|𝒙|^𝟐 ] + c Putting v = 𝑦/𝑥 tan−1 𝒚/𝒙 "= " 𝟏/𝟐 "log" [" " (𝟏+(𝒚/𝒙)^𝟐 )×𝒙^𝟐 ]+𝐜 tan−1 𝑦/𝑥 "= " 1/2 "log" [(𝑥^2 + 𝑦^2)/𝑥^2 ×𝑥^2 ]+c tan−1 𝑦/𝑥 "= " 1/2 "log" [𝑥^2+𝑦^2 ]+c tan−1 𝒚/𝒙 "= " 𝟏/𝟐 "log" [𝒙^𝟐+𝒚^𝟐 ]+𝐜 is the required solution

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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.