Ex 9.3, 5 - Form differential equation: y = ex(a cos x + b sin x) - Formation of Differntial equation when general solution given


  1. Chapter 9 Class 12 Differential Equations
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Ex 9.3, 5 Form a differential equation representing the given family of curves by eliminating arbitrary constants π‘Ž and 𝑏. 𝑦=𝑒^π‘₯ (π‘Ž cos⁑〖π‘₯+𝑏 sin⁑π‘₯ γ€— ) Again Differentiating both sides w.r.t.x 𝑦^β€²β€²=𝑦^β€²+𝑑(𝑒^π‘₯ )/𝑑π‘₯ [βˆ’π‘Ž sin⁑π‘₯+𝑏 cos⁑π‘₯]+𝑒^π‘₯ 𝑑/𝑑π‘₯ [βˆ’π‘Ž sin⁑π‘₯+𝑏 cos⁑π‘₯] 𝑦^β€²β€²=𝑦^β€²+𝒆^𝒙 [βˆ’π’‚ π’”π’Šπ’β‘π’™+𝒃 𝒄𝒐𝒔⁑𝒙]+𝑒^π‘₯ [βˆ’π‘Ž cos⁑π‘₯+𝑏 (βˆ’sin⁑π‘₯)] 𝑦^β€²β€²=𝑦^β€²+γ€–(π’šγ€—^β€²βˆ’ π’š)+𝑒^π‘₯ [βˆ’π‘Ž cos⁑π‘₯βˆ’π‘ sin⁑π‘₯] 𝑦^β€²β€²=2𝑦^β€²βˆ’π‘¦βˆ’π‘’^π‘₯ [π‘Ž cos⁑π‘₯+𝑏 sin⁑π‘₯] 𝑦^β€²β€²=2𝑦^β€²βˆ’π‘¦βˆ’π‘¦ 𝑦^β€²β€²=2𝑦^β€²βˆ’2𝑦 𝑦^β€²β€²βˆ’2𝑦^β€²+2𝑦=0 which is the required differential equation

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