Example 3 Verify that the function 𝑦=𝑎 cos⁡〖𝑥+𝑏 sin⁡〖𝑥, 〗〗 where , 𝑎, 𝑏∈𝐑 is a solution of the differential equation (𝑑^2 𝑦)/(𝑑𝑥^2 )+𝑦=0 𝑦=𝑎 cos⁡〖𝑥+𝑏 sin⁡〖𝑥 〗〗 𝑑𝑦/𝑑𝑥=𝑑/𝑑𝑥 (𝑎 cos⁡〖𝑥+𝑏 sin⁡〖𝑥 〗〗 ) =𝑎 𝑑(cos⁡𝑥 )/𝑑𝑥+𝑏 𝑑(sin⁡𝑥 )/𝑑𝑥 =𝑎(〖−sin〗⁡𝑥 )+𝑏(cos⁡𝑥 ) =−𝑎 𝑠𝑖𝑛𝑥+𝑏 𝑐𝑜𝑠𝑥 (𝑑^2 𝑦)/(𝑑𝑥^2 )=𝑑/𝑑𝑥 (𝑑𝑦/𝑑𝑥) =𝑑/𝑑𝑥 (−𝑎 𝑠𝑖𝑛𝑥+𝑏 𝑐𝑜𝑠𝑥) =−𝑎 𝑑(sin⁡𝑥 )/𝑑𝑥+𝑏 (𝑑(cos⁡𝑥))/𝑑𝑥 =−𝑎(cos⁡𝑥 )+𝑏(−sin⁡𝑥) =−𝑎 cos⁡〖𝑥−𝑏 sin⁡𝑥 〗 Now, we have to verify (𝑑^2 𝑦)/(𝑑𝑥^2 )+𝑦=0 Taking L.H.S (𝑑^2 𝑦)/(𝑑𝑥^2 )+𝑦 =(−𝑎 cos⁡〖𝑥−𝑏 sin⁡𝑥 〗 )+(𝑎 cos⁡〖𝑥+𝑏 sin⁡𝑥 〗 ) =0 = R.H.S ∴ Hence Verified

Next: Example 4 → Ask a doubt

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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.

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