maths.free › Differential Equations

Differential Equations

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

Poučení

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.

Další lekce

Introduction to Differential Equations

Basics of Differential EquationsDirection Fields and Numerical MethodsThe Logistic EquationFirst-Order Linear Equations

Differential Equations

An introduction to differential equationsQualitative behavior of solutions to differential equationsEuler's methodSeparable differential equationsModeling with differential equationsPopulation growth and the logistic equation

Second-Order Differential Equations

Second-Order Linear EquationsNonhomogeneous Linear EquationsApplicationsSeries Solutions of Differential Equations

First order

Differential equationOrdinary differential equationSeparation of variablesIntegrating factorLinear differential equationExact differential equationBernoulli differential equationInitial value problemSlope fieldPicard–Lindelöf theorem

Higher order and systems

Characteristic equation (calculus)Method of undetermined coefficientsVariation of parametersWronskianSystem of differential equationsPhase portraitStability theoryHarmonic oscillator

Transforms and PDEs

Laplace transformFourier transformPartial differential equationHeat equationWave equationLaplace's equationSturm–Liouville theoryGreen's functionNavier–Stokes equationsNumerical methods for ordinary differential equations

Některé z těchto lekcí jsou přizpůsobeny otevřeným textům: Boelkins, Active Calculus, OpenStax Calculus Volume 2, OpenStax Calculus Volume 3, Wikipedia. Každá lekční stránka jmenuje svůj zdroj a licenci.

Zde použité symboly

Ťukněte na jakýkoliv symbol pro úplnou definici, obrázek a co každé písmeno v něm znamená.

Otázky, které se lidé ptají

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.

Máš nějaký problém?
Napište to nebo nakreslete na stránku řešitele: každý krok, obrázek, odpověď skrytou, dokud se nezeptáte. Vyřešit →

Ostatní pobočky

＋ Arithmetic𝑥 Algebra△ Geometry∿ Trigonometryƒ Precalculusσ Statistics & Probability⊢ Discrete Math & Logic∫ Calculus⊞ Linear Algebraℕ Number Theory∇ Multivariable Calculusε Real Analysis⋔ Combinatorics & Graph Theory∈ Set Theory & Logic𝔾 Abstract Algebra𝑃 Probability Theory≈ Numerical Methodsℂ Complex Analysis◎ Topologyμ Measure Theory‖·‖ Functional Analysisκ Differential Geometryπ₁ Algebraic Topology→ Category Theory∂ Partial Differential Equations≋ Fluid Dynamics∞ Frontiers