Misc 2 - Chapter 9 Class 12 Differential Equations - Part 4

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Misc 2 - Chapter 9 Class 12 Differential Equations - Part 5

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Misc 2 - Chapter 9 Class 12 Differential Equations - Part 6

  1. Chapter 9 Class 12 Differential Equations (Term 2)
  2. Serial order wise

Transcript

Misc 2 For each of the exercise given below , verify that the given function (π‘–π‘šπ‘π‘™π‘–π‘π‘–π‘‘ π‘œπ‘Ÿ 𝑒π‘₯𝑝𝑙𝑖𝑐𝑖𝑑) is a solution of the corresponding differential equation . (ii) 𝑦=𝑒^π‘₯ (π‘Ž cos⁑〖π‘₯+𝑏 sin⁑π‘₯ γ€— ) : (𝑑^2 𝑦)/(𝑑π‘₯^2 )βˆ’2 𝑑𝑦/𝑑π‘₯+2𝑦=0 𝑦=𝑒^π‘₯ (π‘Ž cos⁑〖π‘₯+𝑏 sin⁑π‘₯ γ€— ) : Differentiating w.r.t. x 𝑦^β€²=[𝑒^π‘₯ (π‘Ž cos⁑〖π‘₯+𝑏 sin⁑π‘₯ γ€— )]^β€² 𝑦^β€²=γ€–(𝑒〗^π‘₯)β€² (π‘Ž cos⁑〖π‘₯+𝑏 sin⁑π‘₯ γ€— )+𝑒^π‘₯ (π‘Ž cos⁑〖π‘₯+𝑏 sin⁑π‘₯ γ€— )β€² 𝑦^β€²=𝒆^𝒙 (𝒂 𝒄𝒐𝒔⁑〖𝒙+𝒃 π’”π’Šπ’β‘π’™ γ€— )+𝑒^π‘₯ (βˆ’π‘Ž sin⁑〖π‘₯+𝑏 cos⁑π‘₯ γ€— ) Putting 𝑦=𝑒^π‘₯ (π‘Ž π‘π‘œπ‘ β‘γ€–π‘₯+𝑏 𝑠𝑖𝑛⁑π‘₯ γ€— ) : 𝑦^β€²=π’š+𝑒^π‘₯ (βˆ’π‘Ž sin⁑〖π‘₯+𝑏 cos⁑π‘₯ γ€— ) 𝑦^β€²βˆ’π‘¦=𝑒^π‘₯ (βˆ’π‘Ž sin⁑〖π‘₯+𝑏 cos⁑π‘₯ γ€— ) Differentiating again w.r.t x 𝑦^β€²β€²βˆ’π‘¦^β€²=[𝑒^π‘₯ (βˆ’π‘Ž sin⁑〖π‘₯+𝑏 cos⁑π‘₯ γ€— )]^β€² 𝑦^β€²β€²βˆ’π‘¦^β€²=γ€–(𝑒〗^π‘₯)β€²(βˆ’π‘Ž sin⁑〖π‘₯+𝑏 cos⁑π‘₯ γ€— )+𝑒^π‘₯ (βˆ’π‘Ž sin⁑〖π‘₯+𝑏 cos⁑π‘₯ γ€— )β€² 𝑦^β€²β€²βˆ’π‘¦^β€²=𝑒^π‘₯ (βˆ’π‘Ž sin⁑〖π‘₯+𝑏 cos⁑π‘₯ γ€— )+𝑒^π‘₯ (βˆ’π‘Ž cos⁑〖π‘₯βˆ’π‘ si𝑛⁑π‘₯ γ€— ) 𝑦^β€²β€²βˆ’π‘¦^β€²=𝑒^π‘₯ (βˆ’π‘Ž sin⁑〖π‘₯+𝑏 cos⁑π‘₯ γ€— )βˆ’π‘’^π‘₯ (π‘Ž cos⁑〖π‘₯+𝑏 si𝑛⁑π‘₯ γ€— ) Putting 𝑦=𝑒^π‘₯ (π‘Ž π‘π‘œπ‘ β‘γ€–π‘₯+𝑏 𝑠𝑖𝑛⁑π‘₯ γ€— ) 𝑦^β€²β€²βˆ’π‘¦^β€²=𝑒^π‘₯ (βˆ’π‘Ž sin⁑〖π‘₯+𝑏 cos⁑π‘₯ γ€— )βˆ’π‘¦ Putting 𝑦^β€²βˆ’π‘¦=𝑒^π‘₯ (βˆ’π‘Ž 𝑠𝑖𝑛⁑〖π‘₯+𝑏 π‘π‘œπ‘ β‘π‘₯ γ€— ) from (1) …(1) 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 (𝑑^2 𝑦)/(𝑑π‘₯^2 )βˆ’2 𝑑𝑦/𝑑π‘₯+2𝑦=0 Thus, Given Function is a solution of the Differential Equation

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.