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Ex 9.3, 5 - Form differential equation: y = ex(a cos x + b sin x)

Ex 9.3, 5 - Chapter 9 Class 12 Differential Equations - Part 2

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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€— ) Since it has two variables, we will differentiate twice 𝑦=𝑒^π‘₯ (π‘Ž cos⁑〖π‘₯+𝑏 sin⁑π‘₯ γ€— ) Differentiating Both Sides w.r.t. π‘₯ 𝑑𝑦/𝑑π‘₯=𝑑/𝑑π‘₯ [𝑒^π‘₯ (π‘Ž cos⁑π‘₯+𝑏 sin⁑π‘₯ )] 𝑦^β€²=𝑑(𝑒^π‘₯ )/𝑑π‘₯.[π‘Ž cos⁑π‘₯+𝑏 sin⁑π‘₯]+𝑒^π‘₯ 𝑑/𝑑π‘₯ [π‘Ž cos⁑π‘₯+𝑏 sin⁑π‘₯] 𝑦^β€²=𝑒^π‘₯ [π‘Ž cos⁑π‘₯+𝑏 sin⁑π‘₯]+𝑒^π‘₯ [βˆ’π‘Ž sin⁑π‘₯+𝑏 cos⁑π‘₯] 𝑦^β€²=𝑦+𝑒^π‘₯ [βˆ’π‘Ž sin⁑π‘₯+𝑏 cos⁑π‘₯] 𝑦^β€²βˆ’π‘¦=𝑒^π‘₯ [βˆ’π‘Ž sin⁑π‘₯+𝑏 cos⁑π‘₯] …(1) Again Differentiating both sides w.r.t.x 𝑦^β€²β€²βˆ’π‘¦^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€²βˆ’π‘¦^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€²βˆ’π‘¦^β€²=γ€–(π’šγ€—^β€²βˆ’ π’š)+𝑒^π‘₯ [βˆ’π‘Ž cos⁑π‘₯βˆ’π‘ sin⁑π‘₯] 𝑦^β€²β€²βˆ’π‘¦^β€²=𝑦^β€²βˆ’π‘¦βˆ’π’†^𝒙 [𝒂 𝒄𝒐𝒔⁑𝒙+𝒃 π’”π’Šπ’β‘π’™] 𝑦^β€²β€²βˆ’π‘¦^β€²=𝑦^β€²βˆ’π‘¦βˆ’π‘¦ 𝑦^β€²β€²βˆ’π‘¦^β€²=𝑦^β€²βˆ’2𝑦 𝑦^β€²β€²βˆ’π‘¦^β€²βˆ’π‘¦^β€²+2𝑦=0 𝑦^β€²β€²βˆ’2𝑦^β€²+2𝑦=0 which is the required differential equation (From (1)) (Using y = 𝑒^π‘₯ (π‘Ž cos x + b sin x))

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.