Example 20 - Find general solution: x dy/dx + 2y = x2 - Examples

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Example 20 Find the general solution of the differential equation 𝑥 𝑑𝑦﷮𝑑𝑥﷯+2𝑦= 𝑥﷮2﷯ 𝑥≠0﷯ 𝑥 𝑑𝑦﷮𝑑𝑥﷯+2𝑦= 𝑥﷮2﷯ 𝑥 𝑑𝑦﷮𝑥 𝑑𝑥﷯ + 2𝑦﷮𝑥﷯ = 𝑥﷮2﷯﷮𝑥﷯ Dividing both sides by x 𝑑𝑦﷮𝑑𝑥﷯ + 2𝑦﷮𝑥﷯ = x Differential equation is of the form 𝑑𝑦﷮𝑑𝑥﷯+𝑃𝑦=𝑄 where P = 2﷮𝑥﷯ & Q = x IF = 𝑒spdx IF = 𝑒﷮ ﷮﷮ 2﷮𝑥﷯ 𝑑𝑥﷯﷯ IF = 𝑒﷮2 log﷮𝑥﷯﷯ IF = 𝑒﷮ log﷮𝑥﷯﷮2﷯﷯ I.F = x2 Solution of differential equation is y × IF = ﷮﷮ 𝑄×IF﷯𝑑𝑥+𝑐﷯ yx2 = ﷮﷮𝑥× 𝑥﷮2﷯ 𝑑𝑥+𝑐﷯ yx2 = ﷮﷮ 𝑥﷮3﷯𝑑𝑥+𝑐﷯ x2 y = 𝑥﷮4﷯﷮4﷯+𝑐 y = 𝒙﷮𝟐﷯﷮𝟒﷯+𝒄 𝒙﷮−𝟐﷯ is the required general solution

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