First Order System of Equations
Pre-requisite Linear Algebra
Determinants
Determinants are scalar values associated with square matrices, and they provide essential information about the matrix and the linear system it represents. The determinant of a matrix A is denoted as det(A).
2x2 Case
$$\text{det}\begin{bmatrix}
a &b \\
c & d
\end{bmatrix}=ad-bc$$
3x3 Case
$$\begin{bmatrix}
a &b &c\\
d & e &f\\
g & h &i
\end{bmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)$$
Introduction
Definition
A system of n first-order linear differential equations can be written as
$$x_1'=a_{1,1}x_1+a_{1,2}x_2+\dots +a_{1,n}x_n$$
$$x_2'=a_{2,1}x_1+a_{2,2}x_2+\dots+a_{2,n}x_n$$
$$\dots$$
$$x_n'=a_{n,1}x_1+a_{n,2}x_2+\dots+a_{n,n}x_n$$
Represented as a matrix as
$$\mathbf{X}'=A\mathbf{X}$$
The solutions can be found through solving for the eigenvalues and eigenvectors of the matrix.
The eigenvalue is a real or complex number and an eigenvector is a vector such that
$$Av=\lambda v$$
Analytical Solution
Finding Eigenvalues
The eigenvalues can be found by solving
$$\text{det}(A-\lambda I)=0$$
where det is the determinant, A is the coefficient matrix, \(\lambda\) is the eigenvalue, and I is the appropriately sized Identity matrix.
Finding Eigenvectors
Solve for
$$(A-\lambda I)v=0$$
This can be done by Gaussian elimination.
Constructing General Solution
The general solution is given by
$$\mathbf{X}(t)=\sum c_ie^{\lambda_i t}v_i$$
Example
Problem
$$x_1'=2x_1+3x_2$$
$$x_2'=5x_1+4x_2$$
Finding Eigenavlues
$$A=\begin{bmatrix}
2 & 3\\
5 &4
\end{bmatrix}$$
$$\text{det}(A-\lambda I)=0$$
$$\text{det}\begin{bmatrix}
2-\lambda & 3\\
5 &4-\lambda
\end{bmatrix}=0$$
$$(\lambda-2)(\lambda-4)-15=\lambda^2-6\lambda-7$$
$$(\lambda-7)(\lambda+1)=0$$
$$\lambda_1=7$$
$$\lambda_2=-1$$
Computing Eigenvectors
Lambda = 7
$$\begin{bmatrix}
2-7 & 3\\
5 &4-7
\end{bmatrix}
\begin{bmatrix}
v_{1,1}\\
v_{1,2}
\end{bmatrix}=\begin{bmatrix}
0\\
0
\end{bmatrix}$$
$$\begin{bmatrix}
-5 & 3\\
5 &-3
\end{bmatrix}
\begin{bmatrix}
v_{1,1}\\
v_{1,2}
\end{bmatrix}=\begin{bmatrix}
0\\
0
\end{bmatrix}$$
$$-5v_{1,1}+3v_{1,2}=0$$
$$v_{1,1}=\frac{3}{5}v_{1,2}$$
$$v_1=\begin{bmatrix}
3\\
5
\end{bmatrix}$$
Lambda = -1
$$\begin{bmatrix}
2+1 & 3\\
5 &4+1
\end{bmatrix}
\begin{bmatrix}
v_{2,1}\\
v_{2,2}
\end{bmatrix}=\begin{bmatrix}
0\\
0
\end{bmatrix}$$
$$\begin{bmatrix}
3 & 3\\
5 &5
\end{bmatrix}
\begin{bmatrix}
v_{2,1}\\
v_{2,2}
\end{bmatrix}=\begin{bmatrix}
0\\
0
\end{bmatrix}$$
$$v_{2,1}+v_{2,2}=0$$
$$v_{2,1}=-v_{2,2}$$
$$v_2=\begin{bmatrix}
1\\
-1
\end{bmatrix}$$
Solution
$$\mathbf{X}=c_1\begin{bmatrix}
3\\
5
\end{bmatrix}e^{7t}+c_2\begin{bmatrix}
1\\
-1
\end{bmatrix}e^{-t}$$
Exercises
- Calculate the determinant for the following matrices:
- \(\begin{bmatrix}
1 &4 \\
2 & 3
\end{bmatrix}\)
- \(\begin{bmatrix}
0&2 \\
1 & 3
\end{bmatrix}\)
- Solve the following system of equations:
$$x_1'=2x_1+x_2$$
$$x_2'=3x_1+4x_2$$
- Solve the following system of equations:
$$x_1'=2x_1+1x_2$$
$$x_2'=3x_1$$
- Solve the following system of equations:
$$x_1'=x_1-x_2$$
$$x_2'=2x_1-x_2$$
- Solve the following system of equations:
$$x_1'=2x_1-x_3$$
$$x_2'=2x_2-x_3$$
$$x_3'=-x_1+2x_2+2x_3$$