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Differential Equations

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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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

Chapters from Boelkins, Active Calculus

Every section of the book, condensed into a lesson with its own practice problems.

7. 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

Chapters from OpenStax Calculus Volume 3

Every section of the book, condensed into a lesson with its own practice problems.

7. Second-Order Differential Equations

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

Symbols used here

Euler's number
2.71828…, the base whose exponential is its own derivative.

\lim_{x \to a} f(x)

limit
The value f(x) approaches as x approaches a.

f'(x),\ \frac{dy}{dx}

derivative
Instantaneous rate of change; slope of the graph.

\int f(x)\,dx,\ \int_a^b

integral
Antiderivative (indefinite) or signed area from a to b (definite).

y',\ y''

first and second derivative of y
Prime notation for derivatives with respect to x (or t).

C,\ C_1,\ C_2

arbitrary constants
Constants of integration fixed by initial conditions.

\frac{\partial u}{\partial t},\ \nabla^2 u

partial derivative in time, Laplacian
Rate of change in time; sum of second partials (the diffusion operator).

Questions people ask

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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