What quantity changes?
Population, voltage, temperature, position?
A model begins by choosing the variables whose evolution matters.
Side 97
Differential
Equations
A study of systems defined by how they change. Differential equations turn local rules about rates into trajectories through time, connecting calculus to physical, biological and engineered processes.
state→rate→equation→condition→trajectory
06equation lenses
05solution questions
05model patterns
97Side
A differential equation specifies change without directly giving the future.
The solution is a function whose derivatives satisfy the local rule and whose constants are fixed by initial or boundary conditions.
How does change depend on state?
Derivative rule.
The differential equation specifies local motion through state space.
Which derivatives appear?
First, second, higher?
Order determines how much condition information is needed for a unique trajectory.
Where does the system begin?
Initial or boundary data.
The same equation can generate many solutions until conditions select one.
What function satisfies both?
Exact, qualitative, numerical?
Not every useful differential equation has a closed-form solution.
First-order equations model change determined by current state.
Several recurring forms appear across growth, decay, mixing, finance and transport.
Rate proportional to amount.
Growth and decay produce exponential trajectories when proportionality stays constant.
Growth slows near a limit.
State-dependent feedback introduces saturation into otherwise exponential growth.
Variables can be isolated.
Some nonlinear equations become integrable after rearranging state and time terms.
State enters linearly.
Integrating factors convert a broad class into directly solvable form.
Rate depends on state, not explicit time.
Equilibria and direction fields can often reveal behavior without solving explicitly.
External input drives the system.
Time-dependent terms represent interventions, seasonal inputs or changing environments.
Second-order equations describe systems with inertia.
Acceleration depends on forces, allowing oscillation, resonance, damping and transient response.
State
Position alone is not enough.
Second-order dynamics require both position and velocity to determine the future.
Oscillation
Restoring forces create cycles.
Simple harmonic motion produces sinusoidal behavior around equilibrium.
Damping
Energy loss suppresses motion.
Underdamped, critically damped and overdamped systems return differently.
Forcing
External inputs drive response.
Periodic forcing can produce resonance when frequencies align.
Characteristic roots
Algebra reveals dynamic form.
Roots determine whether solutions decay, grow, oscillate or combine those behaviors.
Many real processes require several coupled state variables.
A system of differential equations lets one variable’s rate depend on the others.
| System | State variables | Coupling |
|---|---|---|
| Predator–prey | Two populations | Each changes the other’s growth rate |
| Epidemic | Susceptible, infected, recovered | Contacts transfer population between compartments |
| Electrical circuit | Voltage, current | Components constrain joint evolution |
| Mechanical motion | Position, velocity | Velocity changes position; force changes velocity |
| Chemical kinetics | Concentrations | Reaction rates couple species |
Nonlinearity changes the problem from solving formulas to understanding behavior.
Multiple equilibria, thresholds and sensitivity can emerge when effects are not proportional.
Several resting states can coexist.
Initial conditions can determine which equilibrium the system approaches.
Small changes can switch outcomes.
Nonlinear feedback can create tipping behavior.
Persistent oscillation without external periodic forcing.
The system can settle onto a repeating orbit.
Nearby states can separate rapidly.
Some nonlinear systems amplify tiny initial differences.
Changing a parameter changes qualitative behavior.
Stable states can appear, disappear or lose stability.
Closed forms are exceptional.
Simulation becomes part of understanding rather than merely a convenience.
Differential equations are useful only when the rate law reflects the mechanism well enough.
Modeling moves repeatedly between physical assumptions, mathematical structure and observed behavior.
Choose state
Identify quantities sufficient to describe the evolving system.
Write balance
Translate inflows, outflows, forces or interactions into rate equations.
Set conditions
Specify the state or boundary values that define the particular problem.
Solve or simulate
Use analytic methods where possible and numerical methods where necessary.
Check mechanism
Compare trajectory, units, limiting behavior and data against the intended system.
Elementary Differential Equationsclassical solution methods
Differential Equations and Dynamical Systemsequations as evolving systems
Mathematical Biologymechanistic rate models
Modeling with ODEsstate, rates and initial conditions