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Example 3 - Verify y = a cos x + b sin x is solution of - Gen and Particular Solution

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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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﷮𝑑 𝑥﷮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﷮𝑑 𝑥﷮2﷯﷯= 𝑑﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯ = 𝑑﷮𝑑𝑥﷯ −𝑎 𝑠𝑖𝑛𝑥+𝑏 𝑐𝑜𝑠𝑥﷯ =−𝑎 𝑑 sin﷮𝑥﷯﷯﷮𝑑𝑥﷯+𝑏 𝑑﷮𝑑𝑥﷯ cos﷮𝑥﷯﷯ =−𝑎 cos﷮𝑥+𝑏 − sin﷮𝑥﷯﷯﷯ =−𝑎 cos﷮𝑥+𝑏− sin﷮𝑥﷯﷯ Now , we have to verify 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯+𝑦=0 Taking L.H.S 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯+𝑦 = −𝑎 cos﷮𝑥−𝑏− sin﷮𝑥﷯﷯﷯+ 𝑎 cos﷮𝑥+𝑏 sin﷮𝑥﷯﷯﷯ =0 = R.H.S ∴ Hence Verified

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