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Example 11 - Find particular solution: dy/dx = 4xy2 - Variable separation - Equation given

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Example 11 Find the particular solution of the differential equation 𝑑𝑦﷮𝑑𝑥﷯=4𝑥 𝑦﷮2﷯ given that 𝑦=1 , 𝑤ℎ𝑒𝑛 𝑥=0 Given differential equation , 𝑑𝑦﷮𝑑𝑥﷯=−4𝑥 𝑦﷮2﷯ 𝑑𝑦﷮ 𝑦﷮2﷯﷯ = (−4 x) dx Integrating both sides. ﷮﷮ 𝑑𝑦﷮ 𝑦﷮2﷯﷯﷯ = ﷮﷮−4𝑥 𝑑𝑥﷯ ﷮﷮ 𝑑𝑦﷮ 𝑦﷮2﷯﷯﷯ = −4 ﷮﷮𝑥 𝑑𝑥﷯ 𝑦﷮−2+1﷯﷮−2+1﷯ = −4. 𝑥﷮2﷯﷮2﷯ + c 𝑦﷮−1﷯﷮−1﷯ = −2x2 + c − 1﷮𝑦﷯ = –2x2 + c y = −1﷮−2𝑥2 + 𝑐﷯ y = −1﷮−(2𝑥2 − 𝑐)﷯ y = 1﷮2𝑥2 − 𝑐﷯ Given that at x = 0, y = 1 Putting x = 0, y = 1, in (1) 1 = 1﷮2 0﷯﷮2﷯﷯ − c 1 = 1﷮−𝐶﷯ c = −1 Put c = −1 in (1) y = 1﷮2 𝑥﷮2﷯﷯ −(−1) y = 1﷮2 𝑥﷮2﷯ + 1﷯ Hence, the particular solution of the equation is y = 𝟏﷮𝟐 𝒙﷮𝟐﷯ + 𝟏﷯

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