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Dealing with complex roots and more on independence

Dealing with complex roots and more on independence -...

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Dealing with complex roots and more on independence
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Complex roots When the roots of the auxiliary equation for the differential equation ay’’+by’+cy=0 ar 2 +br+c=0 are complex ( b 2 4ac<0 ) , they come as a complex conjugate pair r= α ±i β In that case we have (complex) solutions y h =C 1 e ( α +i β )t +C 2 e ( α‐ i β )t =e α t (C 1 e i β t +C 2 e i β t )
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Complex roots y h =e α t (C 1 e i β t +C 2 e i β t ) =e α t (C 1 ( cos β t +isin β t) +C 2 ( cos β t isin β t) ) =e α t ((C 1 +C 2 ) cos β t + (iC 1 iC 2 ) sin β t =e α t (D 1 cos β t +D 2 sin β t ) since the C’s were arbitrary constants, we can reformulate and use D’s. This form has the advantage of producing real values when D’s are real.
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Repeated roots In the case with one real root, (b 2 =4ac ) we can seek a second independent solution in the form y=ue rt . In this case, we know that r= b/2a Plugging in to the differential equation we get a(u”+2ru’+r 2 u)e rt +b(u’+ru)e rt +cue rt =0 rearranging we get ((ar 2 +br+c)u+(2ar+b)u’+au’’)e rt =0
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