Hence as long as 0 that is we have complex roots then

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Unformatted text preview: µ eλt sin µt eλt sin µt ￿ ￿ = e2λt λ cos µt sin µt + µ cos2 µt − λ cos µt sin µt + µ sin2 µt ￿ ￿ = µ e2λt cos2 µt + sin2 µt = µ e2λt ￿= 0. Hence as long as µ ￿= 0 – that is, we have complex roots – then {y1 , y2 } does indeed form a fundamental set of solutions. We remark that when the discriminant b2 − 4 a c < 0 that there is not just one fundamental set of solutions. Indeed, we can either choose {y1 , y2 } as above, or we may choose the exponentials {er1 t , er2 t }. The difference is that the exponentials are complex-valued functions, whereas y1 and y2 are real-valued functions. Either choice is fine for mos...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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