Misc 10 - Solve y ex/y dx = (ex/y + y2)dy - Chapter 9 Class 12 - Miscellaneous

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  1. Chapter 9 Class 12 Differential Equations
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Misc 10 Solve the differential equation 𝑦 𝑒﷮ 𝑥﷮𝑦﷯﷯ 𝑑𝑥= 𝑒﷮ 𝑥﷮𝑦﷯﷯+ 𝑦﷮2﷯﷯𝑑𝑦 𝑦≠0﷯ 𝑦 𝑒﷮ 𝑥﷮𝑦﷯﷯ 𝑑𝑥= 𝑒﷮ 𝑥﷮𝑦﷯﷯+ 𝑦﷮2﷯﷯𝑑𝑦 𝑑𝑦﷮𝑑𝑥﷯ = 𝑦 𝑒﷮ 𝑥﷮𝑦﷯﷯﷮𝑥 𝑒﷮ 𝑥﷮𝑦﷯ + 𝑦﷮2﷯﷯﷯ 𝑑𝑥﷮𝑑𝑦﷯ = 𝑥 𝑒﷮ 𝑥﷮𝑦﷯﷯ + 𝑦﷮2﷯﷮ 𝑦﷮ 𝑒﷮ 𝑥﷮𝑦﷯﷯﷯ ﷯ 𝑦 𝑒﷮ 𝑥﷮𝑦﷯﷯ 𝑑𝑥﷮𝑑𝑦﷯ = x 𝑒﷮ 𝑥﷮𝑦﷯﷯ + y2 e﷮ 𝑥﷮𝑦﷯﷯ y 𝑑𝑦 𝑥﷮𝑑𝑦﷯ − x 𝑒﷮ 𝑥﷮𝑦﷯﷯ = y2 e﷮ 𝑥﷮𝑦﷯﷯ y 𝑑𝑥﷮𝑑𝑦﷯ −x﷯ = y2 𝑒﷮ 𝑥﷮𝑦﷯﷯ 𝑦 𝑑𝑥﷮𝑑𝑦﷯ − 𝑥﷮ 𝑦﷮2﷯﷯﷯=1 Let 𝑒﷮ 𝑥﷮𝑦﷯﷯ = z Diff w.r.t. y. 𝑒﷮ 𝑥﷮𝑦﷯﷯ 𝑑﷮𝑑𝑦﷯ 𝑥﷮𝑦﷯﷯= 𝑑𝑧﷮𝑑𝑦﷯ 𝑒﷮ 𝑥﷮𝑦﷯﷯ 𝑦 𝑑𝑥﷮𝑑𝑦﷯ − 𝑥﷮ 𝑦﷮2﷯﷯﷯= 𝑑𝑧﷮𝑑𝑦﷯ From (1) 𝑑𝑧﷮𝑑𝑦﷯ = 1 dz = dy Integrating on both sides ﷮﷮𝑑𝑧﷯ = ﷮﷮𝑑𝑦﷯ z = y + c Putting 𝑒﷮ 𝑥﷮𝑦﷯﷯ = z 𝒆﷮ 𝒙﷮𝒚﷯﷯ = y + c

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.