# Homogeneous Linear Systems

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 5.4 – Differential Operators and the Elimination Method for Systems
• Section 9.4 -- Linear Systems in Nominal Form
• Section 9.5 -- Homogeneous Linear Systems with Constant Coefficients
• Section 9.6 -- Complex Eigenvalues

## Expected Educational Results

• Objective 16–1: I can homogeneous linear systems using eigenvalues and eigenvectors.

## Homogeneous Linear Systems

### Normal Form of a System of $n$ Linear DEs

$\pmb{x}^{\,\prime}(t)=\pmb{A}(t)\pmb{x}(t)+\pmb{f}(t)$

where $\displaystyle \pmb{x}(t)=\left[ \begin{array}{c} x_1(t) \\ x_2(t) \\ \vdots \\ x_n(t) \end{array} \right]$,

$\displaystyle \pmb{f}(t)=\left[ \begin{array}{c} f_1(t) \\ f_2(t) \\ \vdots \\ f_n(t) \end{array} \right]$, and

$\displaystyle \pmb{A}(t)=\left[a_{ij}\right]$ is the $n\times n$ coefficient matrix.

NOTE: Lowercase variables in boldface are vectors; uppercase variables in boldface are matrices.

### Normal Form of a Homogenous System of $n$ Linear DEs

$\pmb{x}^{\,\prime}(t)=\pmb{A}(t)\pmb{x}(t)+\pmb{0}$

otherwise, the system is nonhomogenous.

### Normal Form of a Homogenous System of $n$ Linear DEs with Constant Coefficients

This is the above equation where all elements, $a_{ij}$, of the coefficient matrix $\pmb{A}$ are all constants.

### Solutions to Systems of ODEs

If $\pmb{A}$ is a $n \times n$ coefficient matrix of a system of $n$ linear DEs, then $\displaystyle \text{det}\left(\pmb{A}\pmb{x}-\lambda \pmb{I}\right)=\pmb{0}$ is an $\displaystyle n^{\text{th}}$ degree polynomial in $\lambda$, called the characteristic polynomial.

#### Complex Eigenvalues

Let $\displaystyle \lambda=\alpha\pm\beta i$ be the complex conjugate eigenvalues for the characteristic polynomial corresponding to the complex eigenvector $\displaystyle \vec{v}$. Then the solution to the system of DEs, is

$\pmb{x}(t)=c_1\text{Re}\left(e^{\lambda t}\pmb{v}\right)+c_2\text{Im}\left(e^{\lambda t}\pmb{v}\right)$

where $\text{Re}\left(e^{\lambda t}\pmb{v}\right)$ is the real part of the solution and $\text{Im}\left(e^{\lambda t}\pmb{v}\right)$ is the imaginary part of the solution.

NOTE: Only one eigenvalue and its corresponding eigenvector is needed.

Example 04:

Solve the following system of DEs.

$\begin{array}{ll} x^{\,\prime}_1 &= 2x_1+x_2 \\ x^{\,\prime}_2 &= -x_1+2x_2 \end{array}$

Solution:

1. Rewrite system as: $\pmb{x}^{\,\prime}=\left[\begin{array}{cc} 2 & 1 \\ -1 & 2 \end{array}\right]\pmb{x}$

2. Using technology, the eigenvalues of the coefficient matrix are $\displaystyle \lambda=2\pm i$

​x1(* Store the matrix [2,1; -1,2] *)2A = {{2, 1}, {-1, 2}};3​4(* Display matrix *)5(* To check that we entered the matrix correctly *)6MatrixForm[A]7​8(* Method 1 *)9(* Compute eigenvalues of matrix A *)10Eigenvalues[A]11​12(* Compute eigenvectors of matrix A *)13Eigenvectors[A]14​15(* Method 2 *)16(* Compute eigenvalues and eigenvectors *)17Eigensystem[A]
3. Using technology, the eigenvectors of the coefficient matrix are:

• For $\displaystyle \lambda_1=2+i$, $\displaystyle \pmb{v}_1=\left[\begin{array}{c} -i \\ 1 \end{array}\right]$
• For $\displaystyle \lambda_2=2-i$, $\displaystyle \pmb{v}_2=\left[\begin{array}{c} i \\ 1 \end{array}\right]$
4. Find the fundamental solution set, expand $\displaystyle e^{\lambda t}\pmb{v}$ using Euler's formula (See Question 1 in CPT_11_Complex_Solutions.pdf.): $\displaystyle e^{\lambda t}\pmb{v}=e^{(2+i) t}\left[\begin{array}{c} -i \\ 1 \end{array}\right]=e^{2t}\left(\cos{(t)}+i\sin{(t)}\right)\left[\begin{array}{c} -i \\ 1 \end{array}\right]=e^{2t}\left[\begin{array}{c} -i\cos{(t)+\sin{(t)}} \\ \cos{(t)}+i\sin{(t)} \end{array}\right]=e^{2t}\left[\begin{array}{c} \sin{(t)}-i\cos{(t)} \\ \cos{(t)}+i\sin{(t)} \end{array}\right]$

5. The solution to the system: $\displaystyle \pmb{x}(t)=c_1e^{2t}\left[\begin{array}{c} \sin{(t)} \\ \cos{(t)} \end{array}\right]+c_2e^{2t}\left[\begin{array}{c} -\cos{(t)} \\ \sin{(t)} \end{array}\right]$ which is equivalent to

6. $\displaystyle \begin{array}{c} x_1(t)=c_1e^{2t}\sin{(t)}-c_2e^{2t}\cos{(t)} \\ x_2(t)=c_1e^{2t}\cos{(t)}+c_2e^{2t}\sin{(t)}\end{array}$

#### Investigation 02

For the above example, use the other eigenvalue and corresponding eigenvector and verify the equivalence of the solutions.

#### Investigation 03

Use matrix operations to solve the following systems of DEs:

1. $\displaystyle \begin{array}{ll} x_1^{\,\prime} &= -x_1+9x_2 \\ x_2^{\,\prime} &= -7x_1-3x_2 \end{array}$

2. $\displaystyle \pmb{x}^{\,\prime}=\left[ \begin{array}{cc} 5 & 4 \\ -1 & 5 \end{array} \right]\pmb{x}$

3. $\displaystyle \pmb{x}^{\,\prime}=\left[ \begin{array}{cc} 1 & 2 \\ -2 & 1 \end{array} \right]\pmb{x}$

4. $\displaystyle \pmb{x}^{\,\prime}=\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 2 & 1 & -2 \\ 3 & 2 & 1 \end{array} \right]\pmb{x}$