$y'$Differential Equations

Modelling with Rates

Turn a story into a differential equation and let it predict the future: a cooling cup of coffee, carbon dating, a salty tank, and populations that grow until the world fills up.

Stability and Bifurcations

Learn what solutions do without solving anything. Check when solutions exist and are unique, judge stability from a single derivative, and watch a fishery collapse when the catch crosses a tipping point.

Solving by Computer

Most differential equations have no formula, so we step forward numerically. Take Euler's method apart, measure how its error shrinks, and see why Runge–Kutta, with four slope samples per step, became the workhorse of science.

Exact Equations

Some equations are secretly the derivative of something. Spot exactness with one test, rebuild the hidden potential whose level curves are the solutions, and tame stubborn equations with clever substitutions.

Second-Order Equations

Springs, swings and circuits all obey ay” + by' + cy = 0. Guess an exponential, solve a quadratic, and read the motion off its roots: bouncing, creeping, or exactly in between. Then tune a car's shock absorbers.

Forcing and Resonance

Push a swing at the right moment and it goes higher and higher. Find particular solutions by educated guessing, see the general solution as "natural motion + forced response", and watch resonance and beats happen.

The General Machinery

When guessing fails, there is a method that always works. Combine two basic solutions to hit any starting condition, test independence with the Wronskian, vary the parameters, and build a second solution from a first.

Systems of Equations

Two quantities changing together, like predators and prey or currents in two loops. Draw the phase plane, find the straight-line solutions with eigenvectors, and sort every linear system into nodes, saddles, spirals and centres.

Nonlinear Systems

Foxes and rabbits, swinging pendulums: when the equations are not linear, zoom in on each equilibrium and the linear theory comes back. Find equilibria with nullclines, linearise with the Jacobian, and spot conserved quantities.

Laplace Transforms

A machine that turns calculus into algebra: derivatives become multiplication by s. Transform simple functions, solve an initial-value problem by algebra and partial fractions, and shift between time and frequency.

Switches, Kicks and Convolution

Real forces switch on and off and arrive as sudden kicks. Model them with step functions and the Dirac delta, handle them easily with Laplace transforms, and see how the response to one kick determines the response to anything.

Series Solutions

When no formula works, build the solution one power of x at a time. Find a recurrence for the coefficients, rediscover cosine from y” + y = 0, and find out why a series can stop converging for reasons hidden in the complex plane.

Frobenius and Special Functions

At a singular point the plain power series breaks, so multiply it by x to a power. Solve Cauchy–Euler equations, find the indicial roots, and meet the Bessel function that describes drumheads and ripples in a pond.

Fourier Series

Build a square wave out of smooth sine waves, meet the stubborn Gibbs overshoot, find coefficients by integration, and use the series to solve the heat equation, the problem Fourier was working on when he invented it.