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 $\displaystyle \lambda=2\pm i$

   ​x
   1
   (* Store the matrix [2,1; -1,2] *)
   2
   A = {{2, 1}, {-1, 2}};
   3
   ​
   4
   (* Display matrix *)
   5
   (* To check that we entered the matrix correctly *)
   6
   MatrixForm[A]
   7
   ​
   8
   (* Method 1 *)
   9
   (* Compute eigenvalues of matrix A *)
   10
   Eigenvalues[A]
   11
   ​
   12
   (* Compute eigenvectors of matrix A *)
   13
   Eigenvectors[A]
   14
   ​
   15
   (* Method 2 *)
   16
   (* Compute eigenvalues and eigenvectors *)
   17
   Eigensystem[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. $\displaystyle \pmb{x}^{\,\prime}=\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 2 & 1 & -2 \\ 3 & 2 & 1 \end{array} \right]\pmb{x}$

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Last Modified: Monday, 19 October 2020 8:23 EDT