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


Transcript

Ex 9.5, 6 For each of the differential equation given in Exercises 1 to 12, find the general solution : 𝑥 𝑑𝑦/𝑑𝑥+2𝑦=𝑥^2 𝑙𝑜𝑔𝑥 Step 1 : Convert into 𝑑𝑦/𝑑𝑥 + py = Q 𝑥 𝑑𝑦/𝑑𝑥+2𝑦=𝑥^2 𝑙𝑜𝑔𝑥 Dividing both sides by x 𝒅𝒚/𝒅𝒙 + 𝟐𝒚/𝒙 = x log x Step 2 : Find P and Q Differential equation is of the form 𝑑𝑦/𝑑𝑥+𝑃𝑦=𝑄 where P = 𝟐/𝒙 and Q = x log x Step 3 : Finding integrating factor IF = 𝑒^∫1▒〖𝑝 𝑑𝑥〗epdx IF = 𝑒^∫1▒〖2/𝑥 𝑑𝑥〗 IF = 𝑒^(2∫1▒〖1/𝑥 𝑑𝑥〗) IF = 𝑒^(2 log⁡𝑥 ) IF = 𝑒^log⁡〖𝑥^2 〗 IF = 𝒙^𝟐 Step 4 : Solution of the equation Solution is y (IF) = ∫1▒〖(𝑄×𝐼𝐹)𝑑𝑥+𝑐〗 yx2 = ∫1▒〖𝑥 log⁡〖𝑥×𝑥^2 𝑑𝑥+𝑐〗 〗 yx2 = ∫1▒〖𝒍𝒐𝒈⁡𝒙 𝒙^𝟑 〗 + 𝒄 Integrating by parts with ∫1▒〖𝑓(𝑥) 𝑔(𝑥) 𝑑𝑥=𝑓(𝑥) ∫1▒〖𝑔(𝑥) 𝑑𝑥 −∫1▒〖[𝑓^′ (𝑥) ∫1▒〖𝑔(𝑥) 𝑑𝑥] 𝑑𝑥〗〗〗〗 Take f (x) = sin x & g (x) = 𝑒^2𝑥 yx2 = log x ∫1▒〖𝑥^3 𝑑𝑥−∫1▒[𝑑/𝑑𝑥 log⁡〖𝑥 〗 ∫1▒〖𝑥^3 𝑑𝑥〗] 〗dx yx2 = log x (𝑥^4/4)−∫1▒1/𝑥 (𝑥^4/4)𝑑𝑥+𝑐 yx2 = (𝑥^4 log⁡𝑥)/4 − ∫1▒𝑥^3/4 𝑑𝑥+𝑐 yx2 = (𝑥^4 log⁡𝑥)/4 − 𝑥^4/(4 × 4)+𝑐 yx2 = (𝒙^𝟒 𝒍𝒐𝒈⁡𝒙)/𝟒 − 𝒙^𝟒/𝟏𝟔+𝒄 y = (𝑥^4 log⁡𝑥)/(4𝑥^2 ) − 𝑥^4/(16𝑥^2 ) + 𝐶/𝑥^2 y = (𝑥^2 log⁡〖|𝑥|〗)/4 − 𝑥^2/16 + 𝑐𝑥^(−2) y = 𝒙^𝟐/𝟏𝟔 (4 log |"x" | − 1) + 𝒄𝒙^(−𝟐)

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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.