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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

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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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.