Question

Systems of first-order ODE. Why do you only take one complex eigenvector to find the general solution?

(A question about specific details and solutions for ODEs would need to be provided for options)
A) The correct reason
B) An incorrect reason
C) Not enough information to determine
D) None of the above

Answer 1

In a system of first-order ODEs, when the eigenvalues are complex, they come in conjugate pairs. Taking only one complex eigenvector is sufficient to capture the general behavior of the system. Option A

Step-by-step explanation:

In a system of first-order ordinary differential equations (ODEs), the general solution can be found by finding the eigenvalues and eigenvectors of the coefficient matrix. When the eigenvalues are complex, they come in conjugate pairs, and it is only necessary to find one complex eigenvector.

Let's say we have a 2x2 matrix A with complex eigenvalues λ = a + bi and λ* = a - bi. The corresponding eigenvectors are v = c + di and v* = c - di. It can be shown that the general solution to the system is given by:

x(t) = e^(a+bi) (c + di) + e^(a-bi) (c - di)

Since the complex conjugate pairs in the solution will have the same real part, taking only one complex eigenvector is sufficient to capture the general behavior of the system. Option A.