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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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.