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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Example 19 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€— π’š =𝒆^𝒂𝒙 (π’„πŸ 𝒄𝒐𝒔 𝒃𝒙+ π’„πŸ π’”π’Šπ’β‘π’ƒπ’™) 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‘₯ ) π’š^β€²βˆ’π’‚π’š=𝒃𝒆^𝒂𝒙 (γ€–βˆ’π’„γ€—_𝟏 π’”π’Šπ’β‘π’ƒπ’™+𝒄_𝟐 𝒄𝒐𝒔⁑𝒃𝒙 ) 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.