Characteristic Equations with Complex Conjugate Roots

Fact 1: Euler’s Formula Let θ be any real-valued quantity (i.e. not complex). Then, e iθ = cos(θ) + i sin(θ). This is easily verified with a little complex variables and MacLaurin Series expansion. Fact 2: Theorem about Complex Solutions Suppose that u(x) and v (x) are two real-valued functions of a real variable x, and consider the homogeneous linear ODE an (x) y (n) + an−1 y (n−1) + · · · + a1 (x) y 0 + a0 (x)y = 0 where ai (x) is a real-valued function for 0 ≤ i ≤ n. If y (x) = u(x) + iv (x) is a complex-valued solution to the ODE, then y1 (x) = u(x) AND y2 (x) = v (x) are real-valued solutions to the ODE. This can be verified by just plugging y (x) and its derivatives into the ODE and matching the purely real terms and matching the purely imaginary terms. Complex Roots In the final example from Section 4.4, we had y 00 + y = 0 → r 2 + 1 = 0 → r1 = i, r2 = −i. I y1 (x) = e ix = cos(x) + i sin(x) by Fact 1 I Then, {y1,1 (x) = cos(x), y1,2 (x) = sin(x)} are solutions to the ODE by Fact 2 [easily verified]. I y2 (x) = e −ix = cos(x) + i sin(−x) = cos(x) + i(− sin(x)) I Thus, {y2,1 (x) = cos(x), y2,2 (x) = − sin(x)} are solutions to the ODE. But this is not linearly independent with {cos(x), sin(x)}. I Hence, {cos(x), sin(x)} is a linearly independent set of solutions to the ODE. I General Solution: y (x) = C1 cos(x) + C2 sin(x). Fact 3: Complex Roots to Characteristic Equation Suppose r1 = a + bi and r2 = a − bi are a complex conjugate pair of roots (they always occur in complex conjugate pairs by Algebra) to the characteristic equation of a homogeneous linear ODE with constant coefficients of multiplicity k. Then, { e ax cos(bx), xe ax cos(bx), x 2 e ax cos(bx), . . . , x k−1 e ax cos(bx), e ax sin(bx), xe ax sin(bx), x 2 e ax sin(bx), . . . , x k−1 e ax sin(bx) is a linearly independent set of solutions to the ODE. } Special Note about this Chapter Since we are only dealing with second-order homogeneous linear ODEs with constant coefficients... I Char. Eq: ar 2 + br + c = 0 I Only two roots TOTAL. I Hence if r1 = a + bi and r2 = a − bi are complex conjugate roots, their multiplicity must be 1. In general, they always have the same multiplicity (by Algebra). I General Solution: y (x) = C1 e ax cos(bx) + C2 e ax sin(bx). Example Find a general solution to the following ODE. y 00 + 4y 0 + 8y = 0 Example Find a general solution to the following ODE. y 00 − 2y 0 + 2y = 0 Example Find a general solution to the following ODE. y 00 + 16y = 0 Example Find a general solution to the following ODE. y 00 − 16y = 0 Hint: This is 2 DISTINCT REAL ROOTS, not complex roots.