Ex 9.6, 6 - Find general solution: x dy/dx + 2y = x2 log x - Ex 9.6

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Ex 9.6, 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﷮ ﷮﷮𝑝 𝑑𝑥﷯﷯epdx IF = 𝑒﷮ ﷮﷮ 2﷮𝑥﷯ 𝑑𝑥﷯﷯ IF = 𝑒﷮2 ﷮﷮ 1﷮𝑥﷯ 𝑑𝑥﷯﷯ IF = 𝑒﷮2 log﷮𝑥﷯﷯ IF = 𝑒﷮ log﷮ 𝑥﷮2﷯﷯﷯ IF = 𝑥﷮2﷯ Step 4 : Solution is y (IF) = ﷮﷮ 𝑄×𝐼𝐹﷯𝑑𝑥+𝑐﷯ yx2 = ﷮﷮𝑥 log﷮𝑥× 𝑥﷮2﷯ 𝑑𝑥+𝑐﷯﷯ yx2 = ﷮﷮ log﷮𝑥﷯ 𝑥﷮3﷯﷯ + 𝑐 yx2 = log x ﷮﷮ 𝑥﷮3﷯𝑑𝑥− ﷮﷮ 𝑑﷮𝑑𝑥﷯ log﷮𝑥 ﷯ ﷮﷮ 𝑥﷮3﷯𝑑𝑥﷯﷯﷯﷯dx yx2 = log x 𝑥﷮4﷯﷮4﷯﷯− ﷮﷮ 1﷮𝑥﷯﷯ 𝑥﷮4﷯﷮4﷯﷯𝑑𝑥+𝑐 yx2 = 𝑥﷮4﷯ log﷮𝑥﷯﷮4﷯ − ﷮﷮ 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﷯﷮4 𝑥﷮2﷯﷯ − 𝑥﷮4﷯﷮16 𝑥﷮2﷯﷯ + 𝐶﷮ 𝑥﷮2﷯﷯ y = 𝑥﷮2﷯ log﷮|𝑥|﷯﷮4﷯ − 𝑥﷮2﷯﷮16﷯ + 𝑐 𝑥﷮−2﷯ y = 𝒙﷮𝟐﷯﷮𝟏𝟔﷯ (4 log x﷯ − 1) + 𝒄 𝒙﷮−𝟐﷯

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