Example 24 - Verify that y=c1 eax cos bx + c2 eax sin bx

Example 24 - Chapter 9 Class 12 Differential Equations - Part 2
Example 24 - Chapter 9 Class 12 Differential Equations - Part 3

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


Example 24 Verify that the function 𝑦=𝑐1 𝑒^π‘Žπ‘₯ cos⁑〖𝑏π‘₯+𝑐2 𝑒^π‘Žπ‘₯ sin⁑𝑏π‘₯ γ€— , π‘€β„Žπ‘’π‘Ÿπ‘’ 𝑐1 , 𝑐2 are arbitrary constants is a solution of the differential equation (𝑑^2 𝑦)/(𝑑π‘₯^2 )βˆ’2π‘Ž 𝑑𝑦/𝑑π‘₯+(π‘Ž^2+𝑏^2 )𝑦=0 𝑦=𝑐1 𝑒^π‘Žπ‘₯ cos⁑〖𝑏π‘₯+𝑐2 𝑒^π‘Žπ‘₯ sin⁑〖𝑏π‘₯, γ€— γ€— 𝑦 =𝑒^π‘Žπ‘₯ (𝑐1 π‘π‘œπ‘  𝑏π‘₯+ 𝑐2 sin⁑𝑏π‘₯) Differentiating w.r.t. x 𝑦^β€²=(𝑒^π‘Žπ‘₯ )^β€² (𝑐_1 cos⁑𝑏π‘₯+𝑐_2 sin⁑𝑏π‘₯ )+𝑒^π‘Žπ‘₯ (𝑐_1 cos⁑𝑏π‘₯+𝑐_2 sin⁑𝑏π‘₯ )^β€² 𝑦^β€²=π‘Žπ‘’^π‘Žπ‘₯ (𝑐_1 cos⁑𝑏π‘₯+𝑐_2 sin⁑𝑏π‘₯ )+𝑒^π‘Žπ‘₯ (γ€–βˆ’π‘γ€—_1 𝑏 sin⁑𝑏π‘₯+𝑐_2 𝑏 cos⁑𝑏π‘₯ ) 𝑦^β€²=π‘Žπ‘’^π‘Žπ‘₯ (𝑐_1 cos⁑𝑏π‘₯+𝑐_2 sin⁑𝑏π‘₯ )+𝑏𝑒^π‘Žπ‘₯ (γ€–βˆ’π‘γ€—_1 sin⁑𝑏π‘₯+𝑐_2 cos⁑𝑏π‘₯ ) Putting 𝑦 =𝑒^π‘Žπ‘₯ (𝑐1 π‘π‘œπ‘  𝑏π‘₯+ 𝑐2 sin⁑𝑏π‘₯) 𝑦^β€²=π‘Žπ‘¦+𝑏𝑒^π‘Žπ‘₯ (γ€–βˆ’π‘γ€—_1 sin⁑𝑏π‘₯+𝑐_2 cos⁑𝑏π‘₯ ) 𝑦^β€²βˆ’π‘Žπ‘¦=𝑏𝑒^π‘Žπ‘₯ (γ€–βˆ’π‘γ€—_1 sin⁑𝑏π‘₯+𝑐_2 cos⁑𝑏π‘₯ ) Differentiating again w.r.t x 𝑦^β€²β€²βˆ’π‘Žπ‘¦^β€²=(𝑏𝑒^π‘Žπ‘₯ )^β€² (γ€–βˆ’π‘γ€—_1 sin⁑𝑏π‘₯+𝑐_2 cos⁑𝑏π‘₯ )+𝑏𝑒^π‘Žπ‘₯ (γ€–βˆ’π‘γ€—_1 sin⁑𝑏π‘₯+𝑐_2 cos⁑𝑏π‘₯ )^β€² 𝑦^β€²β€²βˆ’π‘Žπ‘¦^β€²=π‘Žπ‘π‘’^π‘Žπ‘₯ (γ€–βˆ’π‘γ€—_1 sin⁑𝑏π‘₯+𝑐_2 cos⁑𝑏π‘₯ )+𝑏𝑒^π‘Žπ‘₯ (γ€–βˆ’π‘γ€—_1 𝑏 cos⁑𝑏π‘₯βˆ’π‘_2 𝑏 sin⁑𝑏π‘₯ ) 𝑦^β€²β€²βˆ’π‘Žπ‘¦^β€²=π‘Žπ‘π‘’^π‘Žπ‘₯ (γ€–βˆ’π‘γ€—_1 sin⁑𝑏π‘₯+𝑐_2 cos⁑𝑏π‘₯ )βˆ’π‘^2 𝒆^𝒂𝒙 (𝒄_𝟏 𝒄𝒐𝒔⁑𝒃𝒙+𝒄_𝟐 π’”π’Šπ’β‘π’ƒπ’™ ) Putting 𝑦 =𝑒^π‘Žπ‘₯ (𝑐1 π‘π‘œπ‘  𝑏π‘₯+ 𝑐2 sin⁑𝑏π‘₯) 𝑦^β€²β€²βˆ’π‘Žπ‘¦^β€²=π‘Žπ’ƒπ’†^𝒂𝒙 (γ€–βˆ’π’„γ€—_𝟏 π’”π’Šπ’β‘π’ƒπ’™+𝒄_𝟐 𝒄𝒐𝒔⁑𝒃𝒙 )βˆ’π‘^2 𝑦 "Putting" 𝑦^β€²βˆ’π‘Žπ‘¦=𝑏𝑒^π‘Žπ‘₯ (γ€–βˆ’π‘γ€—_1 𝑠𝑖𝑛⁑𝑏π‘₯+𝑐_2 π‘π‘œπ‘ β‘π‘π‘₯ ) 𝑦^β€²β€²βˆ’π‘Žπ‘¦^β€²=π‘Ž(𝑦^β€²βˆ’π‘Žπ‘¦)βˆ’π‘^2 𝑦 𝑦^β€²β€²βˆ’π‘Žπ‘¦^β€²=π‘Žπ‘¦^β€²βˆ’π‘Ž^2 π‘¦βˆ’π‘^2 𝑦 𝑦^β€²β€²βˆ’π‘Žπ‘¦^β€²βˆ’π‘Žπ‘¦^β€²+π‘Ž^2 𝑦+𝑏^2 𝑦=0 π’š^β€²β€²βˆ’πŸπ’‚π’š^β€²+(𝒂^𝟐+𝒃^𝟐)π’š=𝟎 Hence verified

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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.