Non-homogeneous Case

Wednesday, 29 August 2018

1:33 PM

Consider the example ﷐𝑦﷮′′﷯+𝑦=﷐𝑥﷮2﷯+𝑥 Can we find one particular solution ﷐𝑦﷮𝜙﷯ to the equation? Try a solution of the form 𝑦=𝐴﷐𝑥﷮2﷯+𝐵𝑥+𝐶 Then ﷐𝑦﷮′﷯=2𝐴𝑥+𝑏, ﷐𝑦﷮′′﷯=2𝐴 Substituting 𝑦=𝐴﷐𝑥﷮2﷯+𝐵𝑥+𝐶 into the equation 2𝐴+𝐴﷐𝑥﷮2﷯+𝐵𝑥+𝐶=﷐𝑥﷮2﷯+𝑥 Equating coefficients ﷐𝑥﷮2﷯: 𝐴=1 ﷐𝑥﷮1﷯: 𝐵=1 ﷐𝑥﷮0﷯: 2𝐴+𝐶=0 Then 𝐶=−2𝐴=−2 Hence 𝑦=﷐𝑥﷮2﷯+𝑥−2 is a solution to the non-homogeneous equation Are there any other solutions to the given equation? Consider the associated homogeneous equation ﷐𝑦﷮′′﷯+𝑦=0 The characteristic equation is ﷐𝜆﷮2﷯+1=0 ﷐λ+𝑖﷯﷐𝜆−𝑖﷯=0 𝜆=±𝑖 𝜆=0±𝑖 The general solution is 𝑦=﷐𝑒﷮0𝑥﷯﷐𝐶﷐cos﷮𝑖𝑥﷯+𝐷﷐sin﷮𝑖𝑥﷯﷯ General solution is 𝑦=𝐶﷐cos﷮𝑥﷯+𝐷﷐sin﷮𝑥﷯ Ink Drawings Ink Drawings Ink Drawings

 

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