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Example 3 - Verify y = a cos x + b sin x is solution of - Examples

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

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Example 3 Verify that the function 𝑦=π‘Ž cos⁑〖π‘₯+𝑏 sin⁑〖π‘₯, γ€— γ€— where , π‘Ž, π‘βˆˆπ‘ is a solution of the differential equation (𝑑^2 𝑦)/(𝑑π‘₯^2 )+𝑦=0 𝑦=π‘Ž cos⁑〖π‘₯+𝑏 sin⁑〖π‘₯ γ€— γ€— 𝑑𝑦/𝑑π‘₯=𝑑/𝑑π‘₯ (π‘Ž cos⁑〖π‘₯+𝑏 sin⁑〖π‘₯ γ€— γ€— ) =π‘Ž 𝑑(cos⁑π‘₯ )/𝑑π‘₯+𝑏 𝑑(sin⁑π‘₯ )/𝑑π‘₯ =π‘Ž(γ€–βˆ’sin〗⁑π‘₯ )+𝑏(cos⁑π‘₯ ) =βˆ’π‘Ž 𝑠𝑖𝑛π‘₯+𝑏 π‘π‘œπ‘ π‘₯ (𝑑^2 𝑦)/(𝑑π‘₯^2 )=𝑑/𝑑π‘₯ (𝑑𝑦/𝑑π‘₯) =𝑑/𝑑π‘₯ (βˆ’π‘Ž 𝑠𝑖𝑛π‘₯+𝑏 π‘π‘œπ‘ π‘₯) =βˆ’π‘Ž 𝑑(sin⁑π‘₯ )/𝑑π‘₯+𝑏 (𝑑(cos⁑π‘₯))/𝑑π‘₯ =βˆ’π‘Ž(cos⁑π‘₯ )+𝑏(βˆ’sin⁑π‘₯) =βˆ’π‘Ž cos⁑〖π‘₯βˆ’π‘ sin⁑π‘₯ γ€— Now, we have to verify (𝑑^2 𝑦)/(𝑑π‘₯^2 )+𝑦=0 Taking L.H.S (𝑑^2 𝑦)/(𝑑π‘₯^2 )+𝑦 =(βˆ’π‘Ž cos⁑〖π‘₯βˆ’π‘ sin⁑π‘₯ γ€— )+(π‘Ž cos⁑〖π‘₯+𝑏 sin⁑π‘₯ γ€— ) =0 = R.H.S ∴ Hence Verified

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.