Free course · Engineering mathematics

Differential equations, one clear step at a time.

A complete first course on ordinary differential equations written for engineering students. Plain explanations, small-step derivations, original worked examples, and solvers that show every line of the solution for your numbers.

Start with Chapter 1 Try a solver

Example

y^{''} + 2 y^{'} + 26 y = 0

Characteristic equation: $r^{2} + 2 r + 26 = 0$
Discriminant $4 - 104 = - 100 < 0$ , so $r = - 1 \pm 5 i$
$y = e^{- x} (c_{1} cos 5 x + c_{2} sin 5 x)$

Chapters

01Differential Equations and Their Solutions3 sections · quiz 02First-Order Equations with Exact Solutions4 sections · quiz 03Applications of First-Order Equations3 sections · quiz 04Higher-Order Linear Equations: Explicit Methods6 sections · quiz 05Applications of Second-Order Linear Equations with Constant Coefficients6 sections · quiz 06Series Solutions of Linear Differential Equations3 sections · quiz 07Systems of Linear Differential Equations8 sections · quiz 08Approximate Methods for First-Order Equations8 sections · quiz 09The Laplace Transform5 sections · quiz

Solvers that show their work

Type in coefficients or a right-hand side, then step through the solution: characteristic roots with synthetic division, trial forms with the modification rule, integrating factors, Wronskians and more. All arithmetic is exact (fractions, not decimals).

Slope Field (Direction Field) PlotterPlot the slope field of y′ = f(x, y) and click to draw solution curves.Differential Equation Solution CheckerCheck step by step whether a function solves a differential equation.Existence and Uniqueness Theorem CheckerTest the hypotheses of the existence and uniqueness theorem at a point.Exact Differential Equation SolverExactness test, integrating factors μ(x) or μ(y), and the implicit solution F(x, y) = C.Separable Differential Equation SolverSeparate, integrate, recover equilibrium solutions and solve for y.Homogeneous Differential Equation Solver (y = vx)Solve y′ = (ax + by)/(cx + dy) with the substitution y = vx.Linear-fractional equation solvery′ = (a₁x + b₁y + c₁)/(a₂x + b₂y + c₂) by translation or z = a₁x + b₁y.First-Order Linear Differential Equation SolverIntegrating factor method for y′ + P(x)y = Q(x), with initial conditions.

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Small steps

Derivations are written out line by line, so you can see exactly where every term comes from.

Endless practice

Each chapter ends with a quiz that generates four new problems every time, each with a full solution.

Works anywhere

No sign-up and no installation. Everything runs in your browser, on a laptop or a phone.