Solving System of Differential Equations: Eigenvalues and Eigenvectors

Published on May 29, 2025

Linear-Algebra

Find the general solution to a system of differential equations X' = AX, given the eigenvalues and eigenvectors of the matrix A. The solution involves finding linearly independent solutions based on the eigenvalues and eigenvectors, including complex conjugate pairs.

Question

Answer

The problem asks us to find the solutions to the system of differential equations

, given the eigenvalues and eigenvectors of the real

matrix

The given eigenvalues and eigenvectors are:

Eigenvalue with eigenvector .
Eigenvalue with eigenvector .
Eigenvalue with eigenvector .

For a system of linear differential equations

, if

is an eigenvalue and

is its corresponding eigenvector, then

is a solution.

For the real eigenvalue

and eigenvector

, a solution is:

For the complex conjugate eigenvalues

and

, and their corresponding eigenvectors

and

, we can find two real solutions.
Let’s consider the complex solution corresponding to

and

We can write

.
And the eigenvector

can be written as

.
Let

, where

and

Now, substitute these into

The two real solutions are the real and imaginary parts of

The general solution to the system

is a linear combination of these linearly independent solutions:

The final answer is

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Solving System of Differential Equations: Eigenvalues and Eigenvectors

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