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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 e it = 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 e rt = e λt · e iμt = e λt (cos μt + i sin μt ) = e λt cos μt + ie λt sin μt. In fact, because we have the Taylor Series expansion e rt = ∞ X k =0 r k k ! t k = ⇒ d dt e rt = r e rt for any complex number r . Hence we can always make sense of the function y ( t ) = e rt 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 00 + 9 y = 0 . We guess a solution in the form y ( t ) = e rt so that we have the characteristic equation r 2 + 9 = 0 = ⇒ r = ± 3 i. Hence the general solution is the function y ( t ) = a 1 e 3 i + a 2 e- 3 i = a 1 (cos3 t + i sin3 t ) + a 2 (cos3 t- i sin3 t ) = c 1 cos3 t + c 2 sin3 t in terms of the constants c 1 = a 1 + a 2 and c 2 = ia 1- ia 2 . Review. Consider the constant coefficient second order differential equation ay 00 + by + cy = 0 . Recall that it has the associated characteristic equation ar 2 + br + c = 0 . We assume that b 2- 4 ac < 0 so that we have complex roots. We may write these complex roots as r 1 = λ + iμ r 2 = λ- iμ in terms of λ =- b 2 a , μ = p | b 2- 4 ac | 2 a . We saw in the previous lecture that we have the expression e r 1 t = e λt · e iμt = e λt (cos μt + i sin μt ) = e λt cos μt + ie λt sin μt ; e r 2 t = e λt · e- iμt = e λt (cos μt- i sin μt ) = e λt cos μt- ie λt sin μt ....
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- Spring '09
- Real Numbers, Cos, Complex number, ERT