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Differential Equations

Equations whose unknown is a function: growth, oscillation, and how physics is written.

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Core Separable equations dy/dx = f(x)g(y): separate, integrate both sides, solve for y. y' = 2y Core First-order linear equations y′ + P(x)y = Q(x) and the integrating factor that makes it integrable. y' + y = x Core Second-order, constant coefficients ay″ + by′ + cy = 0 and the characteristic equation: oscillation, damping, growth. y'' + y = 0 Advanced Nonhomogeneous equations Driving forces: homogeneous solution plus a particular solution. y'' - 3y' + 2y = e^x Core Modelling with differential equations Growth, decay, cooling, mixing: writing the equation from the sentence. y' = 0.05*y

A differential equation relates a quantity to its own rate of change, which is how every law of physics is stated. The solver names the method (separable, integrating factor, characteristic equation), solves, verifies by substitution, and draws the family of solutions.

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Introduction to Differential Equations

Basics of Differential Equations Direction Fields and Numerical Methods The Logistic Equation First-Order Linear Equations

Differential Equations

An introduction to differential equations Qualitative behavior of solutions to differential equations Euler's method Separable differential equations Modeling with differential equations Population growth and the logistic equation

Second-Order Differential Equations

Second-Order Linear Equations Nonhomogeneous Linear Equations Applications Series Solutions of Differential Equations

First order

Differential equation Ordinary differential equation Separation of variables Integrating factor Linear differential equation Exact differential equation Bernoulli differential equation Initial value problem Slope field Picard–Lindelöf theorem

Higher order and systems

Characteristic equation (calculus)Method of undetermined coefficients Variation of parameters Wronskian System of differential equations Phase portrait Stability theory Harmonic oscillator

Transforms and PDEs

Laplace transform Fourier transform Partial differential equation Heat equation Wave equation Laplace's equation Sturm–Liouville theory Green's function Navier–Stokes equations Numerical methods for ordinary differential equations

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What is a differential equation?

An equation whose unknown is a function, relating it to its own derivatives. "The rate of growth is proportional to the population" is y′ = ky, and solving it means finding y as a function of time.

Why does the solution have arbitrary constants?

Integrating loses information: many functions share the same derivative. An n-th order equation has n constants, fixed by n initial or boundary conditions.

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Differential Equations