Ex 9.2, 8 - Verify y - cos y = x : (y sin y + cos y + x) y' = y - Gen and Particular Solution


  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Ex 9.2, 8 Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation : π‘¦βˆ’cos⁑〖𝑦=π‘₯γ€— : (𝑦 sin⁑〖𝑦+cos⁑〖𝑦+π‘₯γ€— γ€— ) γ€– 𝑦〗^β€²=𝑦 π‘¦βˆ’cos⁑〖𝑦=π‘₯γ€— Differentiating both sides w.r.t. π‘₯ 𝑑/𝑑π‘₯ [π‘¦βˆ’γ€–cos 〗⁑𝑦 ]=𝑑π‘₯/𝑑π‘₯ 𝑑(𝑦)/𝑑π‘₯βˆ’π‘‘[cos 𝑦]/𝑑π‘₯=1 𝑑𝑦/𝑑π‘₯βˆ’(βˆ’sin 𝑦) 𝑑𝑦/𝑑π‘₯=1 𝑑𝑦/𝑑π‘₯+𝑠𝑖𝑛 𝑦 𝑑𝑦/𝑑π‘₯=1" " 𝑑𝑦/𝑑π‘₯ [1+sin 𝑦]=1 𝑑𝑦/𝑑π‘₯=1/(1 + sin 𝑦) Now, we have to verify (𝑦𝑠𝑖𝑛 𝑦+cos 𝑦+π‘₯) 𝑦^β€²=𝑦 Taking L.H.S (𝑦 sin⁑〖𝑦+cos⁑〖𝑦+π‘₯γ€— γ€— ) γ€– 𝑦〗^β€² =[𝑦𝑠𝑖𝑛 𝑦+cos 𝑦+π‘¦βˆ’cos⁑𝑦 ] 𝑦^β€² =[𝑦𝑠𝑖𝑛𝑦+𝑦] 𝑦^β€² =𝑦(1+𝑠𝑖𝑛𝑦) 𝑦^β€² =𝑦(1+𝑠𝑖𝑛𝑦)[1/(1 + sin⁑𝑦 )] =𝑦 = R.H.S Hence Verified

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