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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
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|
in terms of
r2 = λ − i µ
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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- Spring '09
- Real Numbers