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Example 21 - Find general solution: ydx - (x + 2y2)dy = 0 - Solving Linear differential equations - Equation given

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Example 21 Find the general solution of the differential equation 𝑦 𝑑𝑥− 𝑥+2 𝑦﷮2﷯﷯𝑑𝑦=0 𝑦 𝑑𝑥− 𝑥+2 𝑦﷮2﷯﷯𝑑𝑦=0 𝑦 𝑑𝑥= 𝑥+2 𝑦﷮2﷯﷯𝑑𝑦 𝑑𝑦﷮𝑑𝑥﷯ = 𝑦﷮𝑥 + 2 𝑦﷮2﷯﷯ This is not of the form 𝑑𝑦﷮𝑑𝑥﷯+𝑃𝑦=𝑄 ∴ we find 𝑑𝑥﷮𝑑𝑦﷯ 𝑑𝑥﷮𝑑𝑦﷯ = 𝑥 + 2 𝑦﷮2﷯﷮𝑦﷯ 𝑑𝑥﷮𝑑𝑦﷯ = 𝑥 ﷮𝑦﷯ + 2 𝑦﷮2﷯﷮𝑦﷯ 𝑑𝑥﷮𝑑𝑦﷯ − 𝑥 ﷮𝑦﷯ = 2y Differential equation is of the form 𝑑𝑥﷮𝑑𝑦﷯ + P1 x = Q1 where P1 = −1﷮𝑦﷯ & Q1 = 2y Now, IF = 𝑒﷮ ﷮﷮ 𝑝﷮1﷯𝑑𝑦﷯﷯ IF = e﷮ ﷮﷮ −1﷮𝑦﷯𝑑𝑦 ﷯﷯ IF = e﷮− log﷮𝑦﷯﷯ IF = e﷮ log﷮ 1﷮𝑦﷯﷯﷯﷯ IF = 1﷮𝑦﷯ Solution is x(IF) = ﷮﷮ 𝑄×𝐼𝐹﷯𝑑𝑦+𝐶 ﷯ x × 1﷮𝑦﷯= ﷮﷮2𝑦× 1﷮𝑦﷯﷯𝑑𝑦+𝑐 𝑥﷮𝑦﷯ = ﷮﷮2𝑑𝑦+𝑐﷯ 𝑥﷮𝑦﷯ = 2𝑦+𝑐 x = y (2y + c) x = 2y2 + cy

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