# Differential Equations

A differential equation relates a function to its own rates of change.
You rarely get the function handed to you; instead you get a law it must
obey at every instant, and the work is recovering the function from that
law.


_Figure 001 — A direction field for y' = y(1−y), with one solution curve threaded so
its tangent matches the field everywhere.
_

The first thing to read off an equation is its **geometry**.
A first-order law assigns a slope to every point of the plane, and its
solutions trace the curves that stay tangent to that field everywhere, so
you can see their shape long before you can write a formula.


_Figure 002 — A phase portrait: a linear system with complex eigenvalues spirals into a
stable equilibrium.
_

For linear systems the geometry becomes a **phase portrait**.
The eigenvalues of the coefficient matrix decide everything — real and
negative gives decay, imaginary gives rotation, positive real part sends
trajectories away — and the whole flow sorts into nodes, saddles, and
spirals.


Exact methods come next: separation, integrating factors, and
characteristic equations solve the first- and second-order equations
outright, while series and the Laplace transform reach the ones that
resist elementary functions.


_Figure 003 — A damped oscillation e^{−γt}cos ωt, decaying inside its exponential
envelope.
_

The second-order linear equation is the workhorse of physics: a mass on a
spring, a circuit, a swinging pendulum. With damping its solution is an
oscillation trapped inside a shrinking exponential envelope — the balance
of restoring force and dissipation made visible.


_Figure 004 — y' = k y: pure exponential growth for k>0, decay for k<0, from one shared
seed.
_

Underneath all of it sits the simplest equation, y' = k y, whose solution
is a bare exponential. Growth, decay, cooling, and half-lives are all this
one law with a sign chosen; every richer model is measured against it.


_Figure 005 — A predator-prey limit cycle: an isolated closed orbit that nearby
trajectories spiral onto.
_

Most equations that matter are _nonlinear_, and there the goal
shifts from formulas to behaviour — equilibria, stability, limit cycles,
and the onset of chaos, questions you answer qualitatively even when no
closed form exists.


When even that fails you integrate numerically, stepping the solution
forward in small increments; and Fourier methods close the loop, turning
the classical partial differential equations of heat, waves, and
potential back into ordinary ones.


---

## Contents

### 1. Foundations

1. [Models, Direction Fields, and Solution Curves](/differential-equations/foundations/models-and-direction-fields)
2. [Classifying Equations: Order, Linearity, ODE vs. PDE](/differential-equations/foundations/classification-and-terminology)

### 2. First-Order Equations

1. [Linear Equations and Integrating Factors](/differential-equations/first-order/linear-first-order-integrating-factors)
2. [Separable and Exact Equations](/differential-equations/first-order/separable-and-exact)
3. [Modeling with First-Order Equations](/differential-equations/first-order/modeling-first-order)
4. [Autonomous Equations, Phase Lines, and Population Dynamics](/differential-equations/first-order/autonomous-and-population-dynamics)
5. [Existence, Uniqueness, and Euler's Method](/differential-equations/first-order/existence-uniqueness-euler)
6. [First-Order Difference Equations](/differential-equations/first-order/first-order-difference-equations)

### 3. Second-Order Linear Equations

1. [Homogeneous Equations, the Wronskian, and Real Roots](/differential-equations/second-order-linear/homogeneous-constant-coefficients)
2. [Complex Roots, Repeated Roots, and Reduction of Order](/differential-equations/second-order-linear/complex-and-repeated-roots)
3. [Nonhomogeneous Equations: Undetermined Coefficients](/differential-equations/second-order-linear/nonhomogeneous-undetermined-coefficients)
4. [Variation of Parameters](/differential-equations/second-order-linear/variation-of-parameters)
5. [Mechanical and Electrical Vibrations](/differential-equations/second-order-linear/mechanical-electrical-vibrations)
6. [Higher-Order Linear Equations](/differential-equations/second-order-linear/higher-order-linear)

### 4. Series Solutions and Special Functions

1. [Power Series Solutions Near Ordinary Points](/differential-equations/series-solutions/power-series-ordinary-points)
2. [Euler Equations, Regular Singular Points, and Frobenius](/differential-equations/series-solutions/regular-singular-frobenius)
3. [Bessel's Equation, Legendre Polynomials, and Special Functions](/differential-equations/series-solutions/bessel-and-special-functions)

### 5. The Laplace Transform

1. [The Laplace Transform: Definition, Properties, and Solving IVPs](/differential-equations/laplace/laplace-definition-ivps)
2. [Step Functions, Discontinuous Forcing, Impulses, and Convolution](/differential-equations/laplace/step-impulse-convolution)

### 6. Systems of First-Order Linear Equations

1. [Matrices, Linear Systems, and the Eigenvalue Toolkit](/differential-equations/systems/matrices-eigenvalues-review)
2. [Homogeneous Constant-Coefficient Systems and Phase Portraits](/differential-equations/systems/constant-coefficient-systems-phase-portraits)
3. [Repeated Eigenvalues, Fundamental Matrices, and Nonhomogeneous Systems](/differential-equations/systems/repeated-eigenvalues-fundamental-matrices)

### 7. Numerical Methods

1. [Euler, Improved Euler, and Runge–Kutta](/differential-equations/numerical/euler-and-runge-kutta)
2. [Multistep Methods, Systems, and Stability](/differential-equations/numerical/multistep-systems-stability)

### 8. Nonlinear Systems and Stability

1. [The Phase Plane, Critical Points, and Stability](/differential-equations/nonlinear/phase-plane-autonomous-stability)
2. [Locally Linear Systems and Liapunov's Method](/differential-equations/nonlinear/locally-linear-and-liapunov)
3. [Population Models, Limit Cycles, and Chaos](/differential-equations/nonlinear/competing-species-predator-prey-limit-cycles)

### 9. PDEs, Fourier Series, and Boundary Value Problems

1. [Fourier Series and Convergence](/differential-equations/pdes-fourier-bvp/fourier-series)
2. [Separation of Variables: Heat, Wave, and Laplace Equations](/differential-equations/pdes-fourier-bvp/heat-wave-laplace-equations)
3. [Sturm-Liouville Theory](/differential-equations/pdes-fourier-bvp/sturm-liouville)

### 10. Historical Notes and the Calculus of Variations

1. [The Calculus of Variations](/differential-equations/history-variations/calculus-of-variations)
2. [Great Problems and the People Who Solved Them](/differential-equations/history-variations/historical-notes)