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lecture_17 (dragged) - MA 36600 LECTURE NOTES: WEDNESDAY,...

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Unformatted text preview: MA 36600 LECTURE NOTES: WEDNESDAY, FEBRUARY 25 Complex Roots Euler’s Formula. We showed in the previous lecture that eit = cos t + i sin t. This is known as Euler’s Formula. In general, if we write r = λ + iµ in terms of real numbers λ and µ, we have the expression ert = eλt · eiµt = eλt (cos µt + i sin µt) = eλt cos µt + i eλt sin µt. In fact, because we have the Taylor Series expansion ∞ ￿ rk ert = tk =⇒ k! d ￿ rt ￿ e = r ert dt k=0 for any complex number r. Hence we can always make sense of the function y (t) = ert as a solution to a homogeneous linear differential equation with constant coefficients. Example. We explain how to find the general solution to the differential equation y ￿￿ + 9 y = 0. We guess a solution in the form y (t) = ert so that we have the characteristic equation r2 + 9 = 0 Hence the general solution is the function =⇒ r = ±3 i. y (t) = a1 e3i + a2 e−3i = a1 (cos 3t + i sin 3t) + a2 (cos 3t − i sin 3t) = c1 cos 3t + c2 sin 3t in terms of the constants c1 = a1 + a2 and c2 = i a1 − i a2 . Review. Consider the constant coefficient second order differential equation a y ￿￿ + b y ￿ + c y = 0. Recall that it has the associated characteristic equation a r2 + b r + c = 0. We assume that b2 − 4 a c < 0 so that we have complex roots. We may write these complex roots as ￿ r1 = λ + i µ |b2 − 4 a c| b in terms of λ=− , µ= . r2 = λ − i µ 2a 2a We saw in the previous lecture that we have the expression er1 t = eλt · eiµt = eλt (cos µt + i sin µt) = eλt cos µt + i eλt sin µt; . er2 t = eλt · e−iµt = eλt (cos µt − i sin µt) = eλt cos µt − i eλt sin µt Both expressions follow from Euler’s Formula eit = cos t + i sin t. In particular, we may write the general solution to the differential equation as y (t) = a1 er1 t + a2 er2 t = c1 eλt cos µt + c2 eλt sin µt in terms of the constants c1 = a1 + a2 and 1 c2 = i a1 − i a2 . ...
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