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lecture_17 (dragged) 1 - 2 MA 36600 LECTURE NOTES WEDNESDAY...

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2 MA 36600 LECTURE NOTES: WEDNESDAY, FEBRUARY 25 Indeed, we verify that the functions y 1 ( t ) = e λ t cos μt and y 2 ( t ) = e λ t sin μt form a fundamental set of solutions to the di ff erential equation by showing that the Wronskian is a nonzero function. We have the derivatives y 1 ( t ) = e λ t cos μt y 1 ( t ) = λ e λ t cos μt μ e λ t sin μt y 2 ( t ) = e λ t sin μt y 2 ( t ) = λ e λ t sin μt + μ e λ t cos μt This gives the Wronskisn W y 1 , y 2 ( t ) = y 1 ( t ) y 2 ( t ) y 1 ( t ) y 2 ( t ) = e λ t cos μt λ e λ t sin μt + μ e λ t cos μt λ e λ t cos μt μ e λ t sin μt e λ t sin μt = e 2 λ t λ cos μt sin μt + μ cos 2 μt λ cos μt sin μt + μ sin 2 μt = μ e 2 λ t cos 2 μt + sin 2 μt = μ e 2 λ t = 0 . Hence as long as μ = 0 – that is, we have complex roots – then { y 1 , y 2 } does indeed form a fundamental set of solutions. We remark that when the discriminant b 2 4 a c < 0 that there is not just one fundamental set of solutions.
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