Systems of Differential Equations — Theory
1. Linear system form
A two-variable homogeneous linear system of first-order differential equations can be written as:
\[
\begin{cases}
x^{\prime}=a x+b y,\\
y^{\prime}=c x+d y.
\end{cases}
\]
In matrix form:
\[
\begin{aligned}
\mathbf{X}^{\prime}&=A\mathbf{X}\\
\mathbf{X}&=\begin{pmatrix}x\\y\end{pmatrix}\\
A&=\begin{pmatrix}a&b\\c&d\end{pmatrix}
\end{aligned}
\]
2. Eigenvalue idea
The matrix eigenvalue method looks for solutions of the form:
\[
\mathbf{X}(t)=e^{\lambda t}\mathbf{v}
\]
Substitute this into \(\mathbf{X}^{\prime}=A\mathbf{X}\):
\[
\lambda e^{\lambda t}\mathbf{v}
=
A e^{\lambda t}\mathbf{v}
\]
Since \(e^{\lambda t}\neq 0\), this becomes:
\[
A\mathbf{v}=\lambda\mathbf{v}
\]
So the system is solved using eigenvalues and eigenvectors of \(A\).
3. Characteristic equation
Eigenvalues are found from:
\[
\det(A-\lambda I)=0
\]
For:
\[
A=\begin{pmatrix}a&b\\c&d\end{pmatrix}
\]
the characteristic equation is:
\[
\lambda^2-\operatorname{tr}(A)\lambda+\det(A)=0
\]
where:
\[
\begin{aligned}
\operatorname{tr}(A)&=a+d\\
\det(A)&=ad-bc
\end{aligned}
\]
4. Real distinct eigenvalues
If \(A\) has two distinct real eigenvalues \(\lambda_1\) and \(\lambda_2\), with eigenvectors
\(\mathbf{v}_1\) and \(\mathbf{v}_2\), then:
\[
\begin{aligned}
\mathbf{X}(t)
&=
C_1e^{\lambda_1t}\mathbf{v}_1
+
C_2e^{\lambda_2t}\mathbf{v}_2
\end{aligned}
\]
The constants \(C_1\) and \(C_2\) are determined from the initial condition.
5. Complex eigenvalues
If the eigenvalues are complex, they appear in a conjugate pair:
\[
\lambda=\alpha\pm\beta i
\]
The solution contains exponential, sine, and cosine behavior. If a complex eigenvector is:
\[
\mathbf{v}=\mathbf{p}+i\mathbf{q}
\]
then a real solution basis is:
\[
\begin{aligned}
\phi_1(t)
&=
e^{\alpha t}
\left(
\mathbf{p}\cos(\beta t)
-
\mathbf{q}\sin(\beta t)
\right)\\
\phi_2(t)
&=
e^{\alpha t}
\left(
\mathbf{p}\sin(\beta t)
+
\mathbf{q}\cos(\beta t)
\right)
\end{aligned}
\]
Complex eigenvalues produce spiral or center behavior in the phase plane.
6. Repeated eigenvalues
If the repeated eigenvalue \(\lambda\) has two independent eigenvectors, then:
\[
\mathbf{X}(t)
=
C_1e^{\lambda t}\mathbf{v}_1
+
C_2e^{\lambda t}\mathbf{v}_2
\]
If there is only one eigenvector, a generalized eigenvector \(\mathbf{w}\) is used:
\[
(A-\lambda I)\mathbf{w}=\mathbf{v}
\]
The solution becomes:
\[
\begin{aligned}
\mathbf{X}(t)
&=
C_1e^{\lambda t}\mathbf{v}
+
C_2e^{\lambda t}
\left(
t\mathbf{v}
+
\mathbf{w}
\right)
\end{aligned}
\]
7. Applying initial conditions
An initial condition has the form:
\[
\mathbf{X}(t_0)
=
\begin{pmatrix}x_0\\y_0\end{pmatrix}
\]
After writing the general solution:
\[
\mathbf{X}(t)
=
C_1\phi_1(t)
+
C_2\phi_2(t)
\]
substitute \(t=t_0\) and solve:
\[
\begin{aligned}
C_1\phi_1(t_0)
+
C_2\phi_2(t_0)
&=
\begin{pmatrix}x_0\\y_0\end{pmatrix}
\end{aligned}
\]
8. Phase plane
The phase plane plots \(y\) against \(x\). Each point \((x,y)\) represents the current state of the system.
The vector field is:
\[
\begin{pmatrix}x^{\prime}\\y^{\prime}\end{pmatrix}
=
\begin{pmatrix}a x+b y\\c x+d y\end{pmatrix}
\]
The trajectory shows how the state moves through the phase plane as \(t\) changes.
9. Stability classification
10. Example
Consider:
\[
\begin{cases}
x^{\prime}=x+y,\\
y^{\prime}=-2x+3y
\end{cases}
\]
The matrix is:
\[
A=
\begin{pmatrix}
1&1\\
-2&3
\end{pmatrix}
\]
The trace and determinant are:
\[
\begin{aligned}
\operatorname{tr}(A)&=1+3=4\\
\det(A)&=(1)(3)-(1)(-2)=5
\end{aligned}
\]
The characteristic equation is:
\[
\lambda^2-4\lambda+5=0
\]
Therefore:
\[
\lambda=2\pm i
\]
Since the real part is positive, the origin is a spiral source.
11. Trace-determinant interpretation
For a \(2\times2\) system, the trace and determinant give a quick qualitative picture:
\[
\lambda^2-\operatorname{tr}(A)\lambda+\det(A)=0
\]
The discriminant is:
\[
\Delta=\operatorname{tr}(A)^2-4\det(A)
\]
If \(\Delta>0\), the eigenvalues are real. If \(\Delta<0\), the eigenvalues are complex.
If \(\det(A)<0\), the equilibrium is a saddle.
13. Common mistakes
- Forgetting matrix form: always write the system as \(\mathbf{X}^{\prime}=A\mathbf{X}\) before finding eigenvalues.
- Using the wrong characteristic equation: for \(2\times2\) matrices, use \(\lambda^2-\operatorname{tr}(A)\lambda+\det(A)=0\).
- Ignoring eigenvectors: eigenvalues alone do not give the full vector solution.
- Misclassifying complex eigenvalues: the sign of the real part determines spiral sink or spiral source.
- Confusing time curves and phase curves: \(x(t)\) and \(y(t)\) are time plots; the phase plane plots \(y\) versus \(x\).
- Forgetting the initial condition: constants \(C_1\) and \(C_2\) are determined from \(\mathbf{X}(t_0)\).
- Using unsupported LaTeX commands: if \(\boldsymbol{\phi}\) does not render correctly, use the safer notation \(\phi_1(t)\) and \(\phi_2(t)\).
- Ignoring units: graph axes should show both numbered values and units.
