Mathematics · Course 04

Differential Equations

A first university course in ordinary differential equations. Learn to solve the equations that can be solved, to read the ones that cannot from their direction fields and phase portraits, and to run them forward by hand when no formula exists.

y^{'} = f (t, y)

y (t_{0}) = y_{0}

x^{'} = A x

y = C e^{k t}

y^{''} + ω^{2} y = 0

L {f} (s)

e^{A t}

n = 0 \sum \infty a_{n} t^{n}

\frac{d y}{d t}

y_{n + 1} = y_{n} + h f (t_{n}, y_{n})

Course map

10 chapters

01Differential equations and initial value problemsWhat a differential equation asks, what counts as an answer, and how to read the answer off a picture before solving anything. The chapter ends with the one theorem that says an initial value problem has exactly one solution, and the two ways it can fail.
02Separable and autonomous equationsThe first family of equations that can be solved outright, and the first that can be understood without being solved. Separation turns an equation into two integrals; the phase line reads the long run off the sign of the right-hand side alone.
03First-order linear and exact equationsTwo shapes of first-order equation that a method always solves. A linear equation becomes the derivative of a product after one well-chosen multiplication; an exact one is the derivative of a function along a curve, and its solutions are that function's level curves.
04Second-order linear equationsEquations in the second derivative, which a value and a slope determine. Solutions add and scale, so the whole answer is a two-dimensional space; for constant coefficients an exponential guess turns the equation into a quadratic, and its roots — real, repeated or complex — say what every solution does.

05Forced equations and oscillationsWhat a push does to a linear system. A forced equation is one particular solution plus every homogeneous one, found by guessing its shape or by letting the constants vary; then the mass on a spring, free, damped, driven, and driven at its own frequency.

06Linear systems and the phase planeTwo quantities that change together. A linear system is a matrix acting on a vector, its solutions are paths through the phase plane, and the eigenvalues of the matrix decide the whole picture: saddles, nodes, spirals and centres.

07Laplace transformsAn integral that turns differentiation into multiplication. Transform a linear equation, solve the algebra, invert with a table: initial values enter by themselves, and forcing that switches on and off, strikes in an instant or arrives as data is handled by the same three steps.

08Series solutionsEquations whose coefficients vary have no characteristic equation, and their solutions are often no combination of familiar functions. A power series still finds them: the equation fixes the coefficients one by one, the singular points say how far the series reaches, and a power of x in front rescues the method at the mildest singular points.

09Nonlinear systems and stabilitySystems whose rates multiply their unknowns together — pendulums, predators and prey, competing species. They rarely have formula solutions, but their equilibria, the Jacobian at each, and a conserved quantity where one exists give the whole phase portrait, with one case the linearisation cannot decide.

10Numerical methods and course reviewWhen no formula exists, an initial value problem can still be run forward, step by step along the slopes the equation gives. Euler’s method and its error, the better methods of Heun and of Runge and Kutta, how to choose a step, and what numbers can and cannot say — then a review of the whole course.

Mapping the lesson.

Differential Equations

10 chapters