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Misc 12 - Solve differential [e-2x / x - y /root x] dx/dy = 1 - 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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Misc 12 Solve the differential equation 𝑒﷮−2 ﷮𝑥﷯﷯﷮ ﷮𝑥﷯﷯− 𝑦﷮ ﷮𝑥﷯﷯﷯ 𝑑𝑥﷮𝑑𝑦﷯=1 𝑥≠0﷯ 𝑒﷮−2 ﷮𝑥﷯﷯﷮ ﷮𝑥﷯﷯− 𝑦﷮ ﷮𝑥﷯﷯﷯ 𝑑𝑥﷮𝑑𝑦﷯=1 𝑒﷮−2 ﷮𝑥﷯﷯﷮ ﷮𝑥﷯﷯− 𝑦﷮ ﷮𝑥﷯﷯ = 𝑑𝑦﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ + 𝑦﷮ ﷮𝑥﷯﷯ = 𝑒﷮−2 ﷮𝑥﷯﷯﷮ ﷮𝑥﷯﷯ Differential equation is of the form 𝑑𝑦﷮𝑑𝑥﷯ + Py = Q where P = 1﷮ ﷮𝑥﷯﷯ & Q = 𝑒﷮−2 ﷮𝑥﷯﷯﷮ ﷮𝑥﷯﷯ IF. = e ﷮﷮pdx﷯ Finding ﷮﷮𝑃 𝑑𝑥﷯ ﷮﷮𝑃 𝑑𝑥= ﷮﷮ 𝑑𝑥﷮ ﷮𝑥﷯﷯﷯ ﷯ ﷮﷮𝑃 𝑑𝑥= ﷮﷮ 𝑥﷮ −1﷮2﷯﷯𝑑𝑥﷯ ﷯ ﷮﷮𝑃 𝑑𝑥= ﷮﷮ 𝑥 −1﷮2﷯ + 1﷮ −1﷮2﷯ + 1﷯𝑑𝑥﷯ ﷯ ﷮﷮𝑃 𝑑𝑥=2 𝑥﷮ 1﷮2﷯﷯ ﷯= 2 ﷮𝑥﷯ ∴ IF = 𝑒﷮2 ﷮𝑥﷯﷯ Solution is y (IF) = ﷮﷮ 𝑄×𝐼𝐹﷯𝑑𝑥+𝑐﷯ y 𝑒﷮2 ﷮𝑥﷯﷯ = ﷮﷮ 𝑒﷮−2 ﷮𝑥﷯﷯﷮ ﷮𝑥﷯﷯× 𝑒﷮2 ﷮𝑥﷯﷯﷯𝑑𝑥+𝑐﷯ y 𝑒﷮2 ﷮𝑥﷯﷯ = ﷮﷮ 𝑑𝑥﷮ ﷮𝑥﷯﷯+𝑐﷯ y 𝑒﷮2 ﷮𝑥﷯﷯ = ﷮﷮ 1﷮ ﷮𝑥﷯﷯𝑑𝑥+𝑐﷯ y 𝑒﷮2 ﷮𝑥﷯﷯ = ﷮﷮ 𝑥﷮ −1﷮2﷯﷯𝑑𝑥+𝑐﷯ y 𝑒﷮2 ﷮𝑥﷯﷯ = ﷮﷮ 𝑥﷮− 1﷮2﷯ + 1﷯ ﷮− 1﷮2﷯ + 1﷯+𝑐﷯ y 𝑒﷮2 ﷮𝑥﷯﷯ = 2 𝑥﷮ 1﷮2﷯﷯ + C y 𝒆﷮𝟐 ﷮𝒙﷯﷯ = 2 ﷮𝒙﷯ + C

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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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