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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

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.