Ex 9.5, 6 - Chapter 9 Class 12 Differential Equations

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)