Differential Equations & Dynamical Systems

This course contains 44 visible knowledge chapters across eight content-sized units of lengths 4, 7, 5, 9, 5, 4, 6, and 4. If a tank drains faster when it is full, or a spring pulls harder when stretched farther, the law describes change rather than the final quantity. Begin by naming the unknown quantity, its input, units, rate, interval, and whether data are initial or boundary values. Build and verify first-order equations before choosing separable, linear, exact, Bernoulli, qualitative, transform, eigenmode, or numerical methods. Picard iteration constructs the local solution promised by existence theory, and Grönwall’s inequality turns uniqueness and data dependence into a visible error bound. Then construct fundamental solution sets with Wronskians, build missing solutions by reduction of order, solve general forcing by variation of parameters, cross regular singular points with Frobenius reasoning, compare zero spacing with Sturm’s theorem, and accumulate system inputs with Duhamel’s formula. The course continues through boundary eigenvalues, defective systems, frequency response, Poincaré–Bendixson periodic-orbit existence, equilibrium exchanges, Hopf creation of periodic orbits, stability, RK4, and stiff approximation. Every method keeps its hypotheses and first failure boundary visible. The final chapter compares exact structure, qualitative prediction, and controlled numerical error for one damped, driven system.

Before this course: Completed Calculus and Linear Algebra. The course assumes derivatives, definite and improper integrals, elementary series, vectors, matrices, eigenvalues, eigenvectors, and complex-number basics. It introduces modeling, phase portraits, existence–uniqueness, transforms, numerical solvers, and stability locally; no programming language, solver package, physics laboratory, real analysis, or topology is assumed at the opening.