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09 · Subject library
Differential Equations
Equations that describe how quantities evolve, from population models to oscillation.
Start with Autonomous Equations →26 complete · 26 total lessons
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01Autonomous EquationsA phase-line approach to autonomous first-order equations, equilibria, stability, and exact solution curves.Full lesson + sheet02Bernoulli Differential EquationsA rigorous method for Bernoulli equations, including the substitution, excluded cases, and reconstruction checks.Full lesson + sheet03Characteristic EquationA precise guide to characteristic roots for constant-coefficient linear differential equations.Full lesson + sheet04Direction FieldsA rigorous guide to constructing and interpreting slope fields without mistaking them for exact solution curves.Full lesson + sheet05Equilibrium SolutionsA precise treatment of equilibria, stability, linearization, and the distinction between local and global behaviour.Full lesson + sheet06Euler's MethodA careful guide to Euler's numerical method, stepwise computation, error sources, and responsible approximation.Full lesson + sheet07Exact Differential EquationsA rigorous method for exact first-order equations, potential functions, integrating checks, and solution domains.Full lesson + sheet08Existence and UniquenessA precise account of local existence, uniqueness, maximal intervals, and what theorem hypotheses do—and do not—guarantee.Full lesson + sheet09First-Order Linear Differential EquationsA rigorous integrating-factor method for first-order linear equations, including intervals and verification.Full lesson + sheet10Homogeneous Differential EquationsA careful treatment of first-order homogeneous equations $y'=F(y/x)$ and the substitution $v=y/x$.Full lesson + sheet11Initial Value ProblemsA rigorous guide to initial conditions, determining constants, checking intervals, and distinguishing existence from computation.Full lesson + sheet12Introduction to Differential EquationsA rigorous orientation to differential equations, solution concepts, classification, modelling, and verification.Full lesson + sheet13Inverse Laplace TransformA precise method for recovering time-domain functions using tables, algebra, shifting, and convolution.Full lesson + sheet14Laplace TransformA rigorous introduction to the Laplace transform, convergence, derivative rules, and initial value problems.Full lesson + sheet15Mechanical VibrationsA rigorous model of free and forced mass–spring motion, damping regimes, resonance, and physical interpretation.Full lesson + sheet16Mixing ProblemsA mass-balance method for well-stirred tank models, variable volume, units, and physically valid intervals.Full lesson + sheet17Order and LinearityA precise classification guide for order, linearity, homogeneity, autonomy, and systems of differential equations.Full lesson + sheet18Phase PlaneA rigorous introduction to planar autonomous systems, trajectories, equilibria, linearization, and nullclines.Full lesson + sheet19Population ModelsA rigorous comparison of exponential, logistic, harvesting, and data-calibrated population models.Full lesson + sheet20Power Series SolutionsA rigorous coefficient-matching method for local power-series solutions near ordinary points.Full lesson + sheet21Second-Order Linear Differential EquationsA structured treatment of homogeneous and forced second-order linear equations, fundamental solutions, and initial data.Full lesson + sheet22Separable Differential EquationsA rigorous method for separable equations, including equilibrium solutions, implicit forms, and maximal intervals.Full lesson + sheet23Step FunctionsA precise guide to unit-step representations, delayed inputs, piecewise functions, and Laplace transforms.Full lesson + sheet24Systems of Differential EquationsA rigorous introduction to first-order systems, matrix exponentials, eigenmodes, and conversion from higher-order equations.Full lesson + sheet25Undetermined CoefficientsA precise trial-function method for particular solutions of constant-coefficient linear equations.Full lesson + sheet26Variation of ParametersA general particular-solution method for second-order linear equations with variable coefficients.Full lesson + sheet
