# Ex 9.6, 6

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Ex 9.6, 6 For each of the differential equation given in Exercises 1 to 12, find the general solution : 𝑥 𝑑𝑦𝑑𝑥+2𝑦= 𝑥2𝑙𝑜𝑔𝑥 Step 1 : Convert into 𝑑𝑦𝑑𝑥 + py = Q 𝑥 𝑑𝑦𝑑𝑥+2𝑦= 𝑥2𝑙𝑜𝑔𝑥 Dividing both sides by x 𝑑𝑦𝑑𝑥 + 2𝑦𝑥 = x log x Step 2 : Find P and Q Differential equation is of the form 𝑑𝑦𝑑𝑥+𝑃𝑦=𝑄 where P = 2𝑥 and Q = x log x Step 3 : Finding integrating factor IF = 𝑒 𝑝 𝑑𝑥epdx IF = 𝑒 2𝑥 𝑑𝑥 IF = 𝑒2 1𝑥 𝑑𝑥 IF = 𝑒2 log𝑥 IF = 𝑒 log 𝑥2 IF = 𝑥2 Step 4 : Solution is y (IF) = 𝑄×𝐼𝐹𝑑𝑥+𝑐 yx2 = 𝑥 log𝑥× 𝑥2 𝑑𝑥+𝑐 yx2 = log𝑥 𝑥3 + 𝑐 yx2 = log x 𝑥3𝑑𝑥− 𝑑𝑑𝑥 log𝑥 𝑥3𝑑𝑥dx yx2 = log x 𝑥44− 1𝑥 𝑥44𝑑𝑥+𝑐 yx2 = 𝑥4 log𝑥4 − 𝑥34𝑑𝑥+𝑐 yx2 = 𝑥4 log𝑥4 − 𝑥44 × 4+𝑐 yx2 = 𝑥4 log𝑥4 − 𝑥416+𝑐 y = 𝑥4 log𝑥4 𝑥2 − 𝑥416 𝑥2 + 𝐶 𝑥2 y = 𝑥2 log|𝑥|4 − 𝑥216 + 𝑐 𝑥−2 y = 𝒙𝟐𝟏𝟔 (4 log x − 1) + 𝒄 𝒙−𝟐

Chapter 9 Class 12 Differential Equations

Serial order wise

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