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Ex 9.6, 12 Find general solution: (x + 3y2) dy/dx = y - 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, 12 For each of the differential equation find the general solution : 𝑥+3 𝑦﷮2﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯=𝑦 𝑦>0﷯ Step 1 : Put In form 𝑑𝑦﷮𝑑𝑥﷯ + py = Q or 𝑑𝑥﷮𝑑𝑦﷯ + P1x = Q1 𝑥+3 𝑦﷮2﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯=𝑦 𝑑𝑦﷮𝑑𝑥﷯ = 𝑦﷮𝑥+3 𝑦﷮2﷯﷯ This is not of the form 𝑑𝑦﷮𝑑𝑥﷯ + Py = Q ∴ We need to find 𝑑𝑥﷮𝑑𝑦﷯ 𝑑𝑥﷮𝑑𝑦﷯ = 𝑥 + 3 𝑦﷮2﷯﷮𝑦﷯ 𝑑𝑥﷮𝑑𝑦﷯ = 𝑥﷮𝑦﷯ + 3 𝑦﷮2﷯﷮𝑦﷯ Step 2: Find P1 and Q1 Comparing with 𝑑𝑦﷮𝑑𝑥﷯ + P1x = Q1 where P1 = −1﷮𝑦﷯ & Q1 = 3y Step 3 : Finding Integrating factor IF = 𝑒﷮ ﷮﷮ 𝑝﷮1﷯﷯𝑑𝑦﷯ IF = 𝑒﷮ ﷮﷮ −1﷮𝑦﷯﷯𝑑𝑦 ﷯ IF = e−log y IF = 𝑒﷮ log﷮ 𝑦﷮−1﷯﷯﷯ IF = y−1 IF = 1﷮𝑦﷯ Step 4 : Solution is x(IF) = ﷮﷮ 𝑄1×𝐼𝐹﷯𝑑𝑦+𝐶﷯ x 1﷮𝑦﷯﷯= ﷮﷮3𝑦× 1﷮𝑦﷯𝑑𝑦+𝐶﷯ 𝑥﷮𝑦﷯ = 3 ﷮﷮𝑑𝑦+𝐶﷯ 𝑥﷮𝑦﷯ = 3𝑦+𝐶 𝒙 = 𝟑 𝒚﷮𝟐﷯+𝑪y

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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