# Ex 9.5, 6 - Chapter 9 Class 12 Differential Equations

Last updated at April 16, 2024 by Teachoo

Ex 9.5

Ex 9.5, 1
Important

Ex 9.5, 2

Ex 9.5, 3 Important

Ex 9.5, 4

Ex 9.5, 5 Important

Ex 9.5, 6 You are here

Ex 9.5, 7 Important

Ex 9.5, 8 Important

Ex 9.5, 9

Ex 9.5, 10

Ex 9.5, 11

Ex 9.5, 12 Important

Ex 9.5, 13

Ex 9.5, 14 Important

Ex 9.5, 15

Ex 9.5, 16 Important

Ex 9.5, 17 Important

Ex 9.5, 18 (MCQ)

Ex 9.5, 19 (MCQ) Important

Last updated at April 16, 2024 by Teachoo

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 log𝑥 ) 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▒〖𝒍𝒐𝒈𝒙 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 log𝑥)/4 − ∫1▒𝑥^3/4 𝑑𝑥+𝑐 yx2 = (𝑥^4 log𝑥)/4 − 𝑥^4/(4 × 4)+𝑐 yx2 = (𝒙^𝟒 𝒍𝒐𝒈𝒙)/𝟒 − 𝒙^𝟒/𝟏𝟔+𝒄 y = (𝑥^4 log𝑥)/(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)