The Central Idea
A system of differential equations contains two or more differential equations involving multiple unknown functions.
For example,
\[ \boxed{ \begin{aligned} x'&=2x+y,\\ y'&=x-3y. \end{aligned} } \]The rate of change of \(x\) depends on both \(x\) and \(y\), and the rate of change of \(y\) also depends on both variables.
The variables evolve together.
Why Systems Arise
A single differential equation is sufficient when one state variable captures the essential behavior.
Many real systems require several state variables.
Examples include:
- two interacting populations;
- several chemical species;
- position and velocity;
- capital and output;
- voltage and current;
- compartments in an epidemic model;
- multiple interacting inventories;
- coupled temperatures or concentrations.
State Variables
The changing quantities in a dynamic model are often called state variables.
If the state variables are
\[ x(t) \]and
\[ y(t), \]then the system state at time \(t\) is represented by the ordered pair
\[ \boxed{(x(t),y(t))}. \]As time changes, this point moves through the plane.
A General Two-Variable System
A general first-order autonomous system has the form
\[ \boxed{ \begin{aligned} x'&=f(x,y),\\ y'&=g(x,y). \end{aligned} } \]The functions \(f\) and \(g\) determine the instantaneous direction of motion in the state plane.
Vector Form
Define the state vector
\[ \mathbf{x}(t) = \begin{bmatrix} x(t)\\ y(t) \end{bmatrix}. \]Then
\[ \mathbf{x}'(t) = \begin{bmatrix} x'(t)\\ y'(t) \end{bmatrix}. \]A system can therefore be written compactly as
\[ \boxed{ \mathbf{x}'=\mathbf{F}(\mathbf{x}) }. \]Linear Systems
A two-variable linear homogeneous system has the form
\[ \begin{aligned} x'&=ax+by,\\ y'&=cx+dy. \end{aligned} \]In matrix form,
\[ \boxed{ \mathbf{x}'=A\mathbf{x} } \]where
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix}. \]Example: Convert to Matrix Form
Consider
\[ \begin{aligned} x'&=3x+2y,\\ y'&=-x+4y. \end{aligned} \]Then
\[ \boxed{ \begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} 3&2\\ -1&4 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} }. \]Systems and Higher-Order Equations
A higher-order differential equation can often be converted into a first-order system.
Suppose
\[ y''+3y'+2y=0. \]Define
\[ x_1=y \]and
\[ x_2=y'. \]Then
\[ x_1'=x_2. \]Also,
\[ x_2'=y''=-3y'-2y. \]Therefore,
\[ \boxed{ \begin{aligned} x_1'&=x_2,\\ x_2'&=-2x_1-3x_2. \end{aligned} } \]Why First-Order Systems Are Fundamental
Rewriting higher-order equations as first-order systems creates a common framework for:
- analysis;
- matrix methods;
- numerical algorithms;
- phase-plane representations;
- computer simulation;
- control systems;
- multivariable modeling.
Equilibrium Points
An equilibrium point is a state at which every derivative is zero.
For
\[ \begin{aligned} x'&=f(x,y),\\ y'&=g(x,y), \end{aligned} \]equilibria satisfy
\[ \boxed{ f(x,y)=0, \qquad g(x,y)=0 }. \]At an equilibrium, the state remains constant.
Example: Find an Equilibrium
Consider
\[ \begin{aligned} x'&=2x+y,\\ y'&=x-3y. \end{aligned} \]Set
\[ 2x+y=0 \]and
\[ x-3y=0. \]From the second equation,
\[ x=3y. \]Substitute into the first:
\[ 6y+y=0. \]Therefore,
\[ y=0, \qquad x=0. \]The equilibrium is
\[ \boxed{(0,0)}. \]Nonzero Equilibria
Equilibria need not occur only at the origin.
Consider
\[ \begin{aligned} x'&=x(2-x-y),\\ y'&=y(3-x-y). \end{aligned} \]Equilibrium points occur where both right sides vanish.
Because each equation contains factors, several equilibrium combinations may occur.
Nonlinear systems can therefore have multiple equilibria.
Solution Curves in the State Plane
A solution consists of functions
\[ x(t) \]and
\[ y(t). \]The pair
\[ (x(t),y(t)) \]traces a curve in the \(xy\)-plane.
This curve is called a trajectory or orbit.
The Direction Field for a System
At each point \((x,y)\), the system
\[ \begin{aligned} x'&=f(x,y),\\ y'&=g(x,y) \end{aligned} \]assigns the vector
\[ \boxed{ \langle f(x,y),g(x,y)\rangle }. \]This vector tells us the instantaneous direction in which a trajectory moves.
Nullclines
A nullcline is a curve where one component of the derivative vanishes.
The \(x\)-nullcline satisfies
\[ f(x,y)=0. \]Along this curve,
\[ x'=0. \]The \(y\)-nullcline satisfies
\[ g(x,y)=0. \]Along this curve,
\[ y'=0. \]Why Nullclines Are Useful
Nullclines divide the plane into regions where the signs of \(x'\) and \(y'\) may be analyzed.
This helps determine whether trajectories move:
- right or left;
- up or down;
- toward an equilibrium;
- away from an equilibrium.
Qualitative behavior can often be understood without solving the system explicitly.
Solving Linear Systems with Eigenvalues
Consider
\[ \mathbf{x}'=A\mathbf{x}. \]Seek a solution of the form
\[ \mathbf{x}(t)=\mathbf{v}e^{\lambda t}. \]Differentiate:
\[ \mathbf{x}'=\lambda\mathbf{v}e^{\lambda t}. \]Substitute:
\[ \lambda\mathbf{v}e^{\lambda t} = A\mathbf{v}e^{\lambda t}. \]Since \(e^{\lambda t}\neq0\),
\[ A\mathbf{v}=\lambda\mathbf{v}. \]This is the eigenvalue equation.
The Characteristic Equation for a Matrix
Nonzero eigenvectors exist when
\[ (A-\lambda I)\mathbf{v}=0. \]Therefore,
\[ \boxed{ \det(A-\lambda I)=0 }. \]This characteristic equation determines the eigenvalues of the system.
Example: A Diagonal System
Consider
\[ \begin{aligned} x'&=2x,\\ y'&=-3y. \end{aligned} \]The equations are already uncoupled.
Therefore,
\[ x=C_1e^{2t} \]and
\[ y=C_2e^{-3t}. \]In vector form,
\[ \boxed{ \mathbf{x}(t) = C_1 \begin{bmatrix} 1\\ 0 \end{bmatrix} e^{2t} + C_2 \begin{bmatrix} 0\\ 1 \end{bmatrix} e^{-3t} }. \]Distinct Real Eigenvalues
Suppose \(A\) has distinct real eigenvalues
\[ \lambda_1,\lambda_2 \]with eigenvectors
\[ \mathbf{v}_1,\mathbf{v}_2. \]Then
\[ \boxed{ \mathbf{x}(t) = C_1\mathbf{v}_1e^{\lambda_1t} + C_2\mathbf{v}_2e^{\lambda_2t} }. \]Example: Distinct Eigenvalues
Consider
\[ \mathbf{x}' = \begin{bmatrix} 3&1\\ 1&3 \end{bmatrix} \mathbf{x}. \]The characteristic equation is
\[ \det \begin{bmatrix} 3-\lambda&1\\ 1&3-\lambda \end{bmatrix} =0. \]Therefore,
\[ (3-\lambda)^2-1=0. \]So
\[ \lambda=4 \]or
\[ \lambda=2. \]Find the Eigenvectors
For
\[ \lambda=4, \]an eigenvector is
\[ \mathbf{v}_1= \begin{bmatrix} 1\\ 1 \end{bmatrix}. \]For
\[ \lambda=2, \]an eigenvector is
\[ \mathbf{v}_2= \begin{bmatrix} 1\\ -1 \end{bmatrix}. \]Therefore,
\[ \boxed{ \mathbf{x}(t) = C_1 \begin{bmatrix} 1\\ 1 \end{bmatrix} e^{4t} + C_2 \begin{bmatrix} 1\\ -1 \end{bmatrix} e^{2t} }. \]Initial Conditions for Systems
A two-variable system typically uses an initial state such as
\[ x(0)=x_0, \qquad y(0)=y_0. \]In vector form,
\[ \boxed{ \mathbf{x}(0)= \begin{bmatrix} x_0\\ y_0 \end{bmatrix} }. \]These conditions determine the constants in the general solution.
Apply an Initial Condition
Continue with
\[ \mathbf{x}(t) = C_1 \begin{bmatrix} 1\\ 1 \end{bmatrix} e^{4t} + C_2 \begin{bmatrix} 1\\ -1 \end{bmatrix} e^{2t}. \]Suppose
\[ \mathbf{x}(0) = \begin{bmatrix} 3\\ 1 \end{bmatrix}. \]Then
\[ C_1 \begin{bmatrix} 1\\ 1 \end{bmatrix} + C_2 \begin{bmatrix} 1\\ -1 \end{bmatrix} = \begin{bmatrix} 3\\ 1 \end{bmatrix}. \]Complete the Initial-Value Example
The component equations are
\[ C_1+C_2=3 \]and
\[ C_1-C_2=1. \]Add:
\[ 2C_1=4. \]Therefore,
\[ C_1=2. \]Then
\[ C_2=1. \]Thus,
\[ \boxed{ \mathbf{x}(t) = 2 \begin{bmatrix} 1\\ 1 \end{bmatrix} e^{4t} + \begin{bmatrix} 1\\ -1 \end{bmatrix} e^{2t} }. \]Eigenvalues and Stability
Eigenvalues often determine local behavior near an equilibrium.
For a linear system
\[ \mathbf{x}'=A\mathbf{x}, \]the origin is:
- typically stable when all eigenvalues have negative real parts;
- unstable when at least one eigenvalue has positive real part;
- more delicate when eigenvalues have zero real part.
Stable Node
If a two-dimensional linear system has two negative real eigenvalues, then solutions generally approach the origin.
The equilibrium is called a stable node.
The slowest-decaying mode often determines the final direction of approach.
Unstable Node
If both eigenvalues are positive real numbers, trajectories generally move away from the origin.
The equilibrium is an unstable node.
Saddle Point
If one eigenvalue is positive and one is negative, the equilibrium is a saddle point.
Some trajectories approach the equilibrium along the stable eigendirection, while others move away along the unstable eigendirection.
Saddle points are unstable overall.
Complex Eigenvalues
If the eigenvalues are
\[ \lambda=\alpha\pm\beta i, \]then trajectories often rotate or spiral.
If
\[ \alpha<0, \]the spiral tends inward.
If
\[ \alpha>0, \]the spiral tends outward.
If
\[ \alpha=0, \]a linear system may produce closed or neutral oscillatory behavior.
Trace and Determinant
For
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix}, \]the trace is
\[ \boxed{\operatorname{tr}(A)=a+d} \]and the determinant is
\[ \boxed{\det(A)=ad-bc}. \]For a \(2\times2\) matrix, the characteristic polynomial can be written
\[ \boxed{ \lambda^2-\operatorname{tr}(A)\lambda+\det(A)=0 }. \]Why Trace and Determinant Matter
The trace equals the sum of the eigenvalues:
\[ \lambda_1+\lambda_2=\operatorname{tr}(A). \]The determinant equals their product:
\[ \lambda_1\lambda_2=\det(A). \]These quantities provide a compact way to classify many two-dimensional linear systems.
Nonhomogeneous Linear Systems
A forced linear system has the form
\[ \boxed{ \mathbf{x}'=A\mathbf{x}+\mathbf{g}(t) }. \]The solution generally consists of:
\[ \boxed{ \mathbf{x} = \mathbf{x}_h+\mathbf{x}_p }. \]The homogeneous part describes internal system behavior, while the particular part describes the response to external input.
Constant Inputs
Suppose
\[ \mathbf{x}'=A\mathbf{x}+\mathbf{b}, \]where \(\mathbf{b}\) is constant.
A constant equilibrium \(\mathbf{x}_*\) satisfies
\[ 0=A\mathbf{x}_*+\mathbf{b}. \]If \(A\) is invertible,
\[ \boxed{ \mathbf{x}_*=-A^{-1}\mathbf{b} }. \]Shifting an Equilibrium to the Origin
If a system has equilibrium \(\mathbf{x}_*\), define
\[ \mathbf{u}=\mathbf{x}-\mathbf{x}_*. \]The shifted variable measures displacement from equilibrium.
For many linear systems with constant input, this transforms the problem into
\[ \boxed{ \mathbf{u}'=A\mathbf{u} }. \]The same eigenvalue methods then apply.
Interacting Population Models
Systems are essential in population dynamics.
A simplified interacting model might be
\[ \begin{aligned} x'&=x(a-bx-cy),\\ y'&=y(d-ex-fy). \end{aligned} \]Each population's growth rate depends on both its own size and the size of the other population.
Predator-Prey Structure
A classical predator-prey model has the form
\[ \begin{aligned} x'&=ax-bxy,\\ y'&=-cy+dxy. \end{aligned} \]Here:
- \(x\) may represent prey;
- \(y\) may represent predators;
- the product \(xy\) represents interaction between the populations.
This is a nonlinear system.
Compartment Models
Systems also arise when a total quantity is divided among compartments.
For example, a simple transfer model may be
\[ \begin{aligned} x'&=-ax+by,\\ y'&=ax-by. \end{aligned} \]Material leaving one compartment enters the other.
Adding the equations gives
\[ x'+y'=0. \]Therefore,
\[ x+y=C. \]The total amount is conserved.
Conservation Laws
A system may contain quantities that remain constant even though individual state variables change.
If
\[ x'+y'=0, \]then
\[ \frac{d}{dt}(x+y)=0. \]Hence
\[ \boxed{x+y=C}. \]Recognizing conserved quantities can simplify a system substantially.
Systems in Mechanics
A second-order mechanical equation
\[ mx''+cx'+kx=F(t) \]can be written as a first-order system by defining
\[ v=x'. \]Then
\[ \boxed{ \begin{aligned} x'&=v,\\ v'&=\frac{1}{m}\left(F(t)-cv-kx\right). \end{aligned} } \]The state variables are position and velocity.
Systems in Control and Engineering
Engineering models often use a state-space form
\[ \boxed{ \mathbf{x}'=A\mathbf{x}+B\mathbf{u} } \]with an output equation
\[ \boxed{ \mathbf{y}=C\mathbf{x}+D\mathbf{u} }. \]Here \(\mathbf{u}\) represents inputs and \(\mathbf{y}\) represents measured outputs.
This is a direct extension of the linear-system framework.
Numerical Solution of Systems
Euler's method extends naturally to systems.
For
\[ \mathbf{x}'=\mathbf{F}(t,\mathbf{x}), \]the vector Euler update is
\[ \boxed{ \mathbf{x}_{n+1} = \mathbf{x}_n + h\mathbf{F}(t_n,\mathbf{x}_n) }. \]Every component is updated using the same time step.
Example: One Euler Step for a System
Suppose
\[ \begin{aligned} x'&=x+y,\\ y'&=x-y, \end{aligned} \]with
\[ x(0)=1, \qquad y(0)=2. \]Use
\[ h=0.1. \]At \((1,2)\),
\[ x'=3 \]and
\[ y'=-1. \]Therefore,
\[ x_1=1+0.1(3)=1.3 \]and
\[ y_1=2+0.1(-1)=1.9. \]Thus,
\[ \boxed{(x(0.1),y(0.1))\approx(1.3,1.9)}. \]Common Error: Solving Each Equation Independently
In a coupled system,
\[ x' \]may depend on \(y\), and
\[ y' \]may depend on \(x\).
The equations cannot generally be solved independently.
Their interaction is the defining feature of the system.
Common Error: Confusing the State Plane with Time
A phase-plane trajectory plots
\[ y(t) \]against
\[ x(t), \]not against time.
Time determines how the point moves along the trajectory but is not usually drawn as an axis in the phase plane.
Common Error: Forgetting to Find All Equilibria
Equilibria require
\[ x'=0 \]and
\[ y'=0 \]simultaneously.
Solving only one equation identifies a nullcline, not necessarily an equilibrium point.
Common Error: Using Eigenvalues Without Eigenvectors
Eigenvalues determine exponential rates.
Eigenvectors determine the associated directions in state space.
For distinct eigenvalues, the solution modes are
\[ \boxed{ \mathbf{v}_ie^{\lambda_it} }. \]Both pieces are required.
Common Error: Ignoring the Sign of Real Parts
For complex eigenvalues
\[ \alpha\pm\beta i, \]the imaginary part controls rotational or oscillatory behavior.
The real part \(\alpha\) controls exponential growth or decay.
Ignoring \(\alpha\) can lead to an incorrect stability conclusion.
Common Error: Assuming Every Nonlinear System Has an Explicit Formula
Many nonlinear systems cannot be solved in elementary closed form.
In those cases, useful analysis may include:
- equilibrium calculations;
- nullclines;
- phase portraits;
- linearization;
- numerical simulation;
- conservation laws.
A Reliable Systems Workflow
- Identify the state variables.
- Write one differential equation for each state variable.
- Determine whether the system is linear or nonlinear.
- Write the system in vector or matrix form when useful.
- Find equilibrium points.
- Use nullclines and signs for qualitative analysis.
- For linear systems, compute eigenvalues and eigenvectors.
- Build the general solution from independent modes.
- Apply the initial state.
- Interpret stability from eigenvalues.
- Use numerical methods when exact formulas are unavailable.
- Check units, conservation laws, and model assumptions.
Example: Full Linear-System Workflow
Solve
\[ \mathbf{x}' = \begin{bmatrix} 2&1\\ 1&2 \end{bmatrix} \mathbf{x}. \]The characteristic equation is
\[ \det \begin{bmatrix} 2-\lambda&1\\ 1&2-\lambda \end{bmatrix} =0. \]Therefore,
\[ (2-\lambda)^2-1=0. \]So
\[ \lambda=3 \]or
\[ \lambda=1. \]Complete the Full Linear-System Workflow
For
\[ \lambda=3, \]an eigenvector is
\[ \mathbf{v}_1= \begin{bmatrix} 1\\ 1 \end{bmatrix}. \]For
\[ \lambda=1, \]an eigenvector is
\[ \mathbf{v}_2= \begin{bmatrix} 1\\ -1 \end{bmatrix}. \]Therefore,
\[ \boxed{ \mathbf{x}(t) = C_1 \begin{bmatrix} 1\\ 1 \end{bmatrix} e^{3t} + C_2 \begin{bmatrix} 1\\ -1 \end{bmatrix} e^{t} }. \]Since both eigenvalues are positive, the origin is unstable.
Practice 1: Matrix Form
Write
\[ \begin{aligned} x'&=4x-y,\\ y'&=2x+3y \end{aligned} \]in matrix form.
Practice 2: Find the Equilibrium
Find the equilibrium of
\[ \begin{aligned} x'&=x+y,\\ y'&=2x-y. \end{aligned} \]Practice 3: Convert a Second-Order Equation
Rewrite
\[ y''+4y'+5y=0 \]as a first-order system.
Practice 4: Eigenvalues
Find the eigenvalues of
\[ A= \begin{bmatrix} 5&0\\ 0&-2 \end{bmatrix}. \]Practice 5: Classify Stability
A linear system has eigenvalues
\[ -2 \]and
\[ -5. \]What type of equilibrium behavior is expected at the origin?
Practice 6: Saddle Point
A system has eigenvalues
\[ 3 \]and
\[ -1. \]Classify the origin.
Practice 7: Complex Eigenvalues
A system has eigenvalues
\[ -1\pm4i. \]Describe the expected qualitative behavior.
Practice 8: One Euler Step
For
\[ \begin{aligned} x'&=2x+y,\\ y'&=x-y, \end{aligned} \]with
\[ x(0)=1, \qquad y(0)=1, \]use Euler's method with
\[ h=0.1 \]to estimate the state at \(t=0.1\).
Practice 9: Conservation
For
\[ \begin{aligned} x'&=-3x+2y,\\ y'&=3x-2y, \end{aligned} \]show that \(x+y\) is constant.
Practice 10: Trace and Determinant
For
\[ A= \begin{bmatrix} 2&1\\ 3&4 \end{bmatrix}, \]find the trace and determinant.
Practice Answers
Practice 1
The matrix form is
\[ \boxed{ \begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} 4&-1\\ 2&3 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} }. \]Practice 2
Set
\[ x+y=0 \]and
\[ 2x-y=0. \]From the first,
\[ y=-x. \]Substitute:
\[ 2x+x=0. \]Therefore,
\[ x=0, \qquad y=0. \]The equilibrium is
\[ \boxed{(0,0)}. \]Practice 3
Let
\[ x_1=y, \qquad x_2=y'. \]Then
\[ x_1'=x_2. \]Also,
\[ x_2'=y''=-4y'-5y. \]Therefore,
\[ \boxed{ \begin{aligned} x_1'&=x_2,\\ x_2'&=-5x_1-4x_2. \end{aligned} } \]Practice 4
Because the matrix is diagonal, the eigenvalues are the diagonal entries:
\[ \boxed{\lambda_1=5,\qquad\lambda_2=-2}. \]Practice 5
Both eigenvalues are real and negative.
Therefore, trajectories generally approach the origin.
The origin is a
\[ \boxed{\text{stable node}}. \]Practice 6
One eigenvalue is positive and one is negative.
Therefore, the origin is a
\[ \boxed{\text{saddle point}}. \]Practice 7
The negative real part causes decay, while the imaginary part produces rotation.
Therefore, trajectories typically
\[ \boxed{\text{spiral inward toward the equilibrium}}. \]Practice 8
At
\[ (x,y)=(1,1), \]we have
\[ x'=2(1)+1=3 \]and
\[ y'=1-1=0. \]Therefore,
\[ x_1=1+0.1(3)=1.3 \]and
\[ y_1=1+0.1(0)=1. \]Thus,
\[ \boxed{(x(0.1),y(0.1))\approx(1.3,1)}. \]Practice 9
Add the equations:
\[ x'+y' = (-3x+2y)+(3x-2y) = 0. \]Therefore,
\[ \frac{d}{dt}(x+y)=0. \]Hence
\[ \boxed{x+y=C}. \]Practice 10
The trace is
\[ 2+4=6. \]The determinant is
\[ 2(4)-1(3)=5. \]Therefore,
\[ \boxed{ \operatorname{tr}(A)=6, \qquad \det(A)=5 }. \]What to Remember
A system of differential equations models several interacting state variables at once.
A linear homogeneous system can be written
\[ \boxed{ \mathbf{x}'=A\mathbf{x} }. \]- equilibria occur where every derivative is zero;
- trajectories describe motion through state space;
- nullclines help organize qualitative behavior;
- higher-order equations can be rewritten as first-order systems;
- eigenvalues determine exponential modes and stability;
- eigenvectors determine preferred directions in state space;
- real eigenvalues often produce nodes or saddles;
- complex eigenvalues often produce rotational or spiral behavior;
- numerical methods extend naturally to vector-valued systems.
The central structural idea is
\[ \boxed{ \text{state vector} \longrightarrow \text{rate vector} \longrightarrow \text{trajectory} }. \]Where We Go Next
Differential equations describe families of possible trajectories. To select a specific solution, we usually impose additional information about the system.
The next lesson, Initial and Boundary Conditions, develops how starting states, endpoint constraints, and boundary data determine particular solutions and affect existence, uniqueness, and model interpretation.