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# 3.3 - Math 334 Lecture#12 3.3 Complex Roots Recall that y =...

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Math 334 Lecture #12 § 3.3: Complex Roots Recall that y = e rt is a solution of the linear homogeneous second order ODE with constant coefficients, ay + by + cy = 0 , when r is a root of the characteristic equation ar 2 + br + c = 0. The roots of the characteristic equation are complex conjugates when the discriminant, b 2 - 4 ac , is negative: r = - b ± b 2 - 4 ac 2 a = - b 2 a ± i 4 ac - b 2 2 a . To simplify the notation, let λ = - b/ 2 a and μ = 4 ac - b 2 / 2 a . The two solutions of corresponding to the complex conjugate roots λ ± are ˆ y 1 = exp[( λ + ) t ] , ˆ y 2 = exp[( λ - ) t ] . [What is the exponential of a complex number?] Euler’s Formula: exp { iμt } = cos μt + i sin μt . [This formula can be derived by replacing x in the power (Taylor) series expansion for e x by iμt , using i 2 = - 1, i 3 = - i , i 4 = 1, then collecting the terms without i together (this gives cos μt ), and the terms with i together (this gives i sin μt ).] By Euler’s Formula, the solutions of the ODE are ˆ y 1 = exp[( λ + ) t ] = exp[ λt + iμt ] = exp { λt } exp { iμt } = e λt [cos μt + i sin μt ] ,

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