Differential Equations
Equations that describe change.
Nine units. The bridge from calculus to the sciences — every physical law is one.
Unit 1 · First look
Modeling with ODEs
What a differential equation says, order, initial conditions, qualitative behavior.
Unit 2 · SeparableFirst-order equations
Separable, linear, exact, integrating factor, Bernoulli.
Unit 3 · ModelsApplications of first order
Growth, decay, cooling, mixing, RC circuits, the logistic equation.
Unit 4 · LinearSecond-order linear ODEs
Homogeneous, characteristic equation, complex roots, undetermined coefficients.
Unit 5 · ResonanceForced oscillations
Harmonic forcing, resonance, beats, damped driven motion.
Unit 6 · OperatorLaplace transforms
Definition, transforms of common functions, solving initial-value problems.
Unit 7 · VectorsSystems of ODEs
Matrix form, phase plane, eigenvalues as the tool for classification.
Unit 8 · SeriesPower-series methods
Solutions near ordinary points, Frobenius, Legendre and Bessel.
Unit 9 · ContinuousPartial differential equations
Separation of variables, heat, wave, and Laplace equations — and a taste of Fourier.