University:
Georgia Southern University
Course:
MATH 3230 | Ordinary Differential Equations
Academic year:

2024
Views:

330

Pages:

4
Author:

Jalen W.

Algorithm: General Solution (Homogeneous Version) 1. Find the characteristic equation. 2. Find the roots to the characteristic equation. 2.1 If r is a real root of multiplicity k, then e rx , xe rx , . . . , x k−1 e rx are k linearly independent solutions to the ODE. 2.2 If r = a + bi is a complex root of multiplicity k (r = a − bi is also a complex root of multiplicity k), then e ax sin(bx), xe ax sin(bx), . . . , x k−1 e ax sin(bx) e ax cos(bx), xe ax cos(bx), . . . , x k−1 e ax cos(bx) are 2k linearly independent solutions to the ODE. 3. Finally, y (x) is a linear combination of your solutions. Example Find the general solution to the ODE. y (6) + y (4) = 0 Example Suppose the characteristic equation to a homogeneous linear ODE with constant coefficients is given by r 2 (r + 1)3 (r 2 − 2r + 2)2 = 0. Find the general solution to the ODE. Example Suppose the roots and multiplicities to the characteristic equation of some homogeneous linear ODE with constant coefficients is given below. r1 = 2, k1 = 3 r2 = 2 + i, k2 = 1 r3 = 2 − i, k3 = 1 r4 = 2i, k4 = 2 r5 = −2i, k5 = 2 Find the general solution to the ODE.

Higher-order Homogeneous Linear ODEs

Higher-order Homogeneous Linear ODEs - Page 3

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