Ordinary Differential Equations

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• What a differential equation is ~12 min
• The order of a differential equation ~8 min
• Linear versus nonlinear equations ~10 min
• Explicit versus implicit solutions ~10 min
• Autonomous versus nonautonomous equations ~8 min
• Verifying a solution ~8 min
• General and particular solutions, families of solutions ~10 min
• Initial-value problems ~8 min
• Existence and uniqueness ~12 min
• Boundary-value problems, a first look ~10 min

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• Separable equations ~12 min
• First-order linear equations and integrating factors ~10 min
• Applications of the integrating-factor method ~10 min
• Differential form and the exactness condition ~12 min
• Solving exact equations ~12 min
• Integrating factors for non-exact equations ~12 min
• The Bernoulli equation ~12 min
• Homogeneous first-order equations ~12 min
• Other useful substitutions ~10 min
• Choosing a solution method ~10 min

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• Slope fields ~10 min
• Isoclines ~9 min
• Phase lines and equilibria ~10 min
• Stability: stable, unstable, semistable ~9 min
• The logistic equation, revisited ~9 min
• Logistic growth with harvesting ~11 min
• A parameter that moves an equilibrium ~8 min
• Bifurcation of a 1-D autonomous family ~10 min
• Long-term behavior without solving ~9 min
• Reading qualitative behavior from a model ~9 min

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• Why a closed-form solution isn't always available ~8 min
• Euler's method ~9 min
• Local versus global error ~9 min
• Heun's method (improved Euler) ~10 min
• Runge–Kutta methods, RK2 ~9 min
• RK4 ~10 min
• Step size and accuracy trade-offs ~9 min
• Reading order of accuracy off an error plot ~9 min
• Stiffness, informally ~9 min

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• Exponential growth and decay ~9 min
• Newton's law of cooling ~9 min
• Mixing problems and two-tank systems ~12 min
• Pharmacokinetics: a drug-elimination model ~10 min
• RC circuits ~10 min
• Chemical reaction rates ~10 min
• Terminal velocity with air resistance ~9 min
• Compartment models ~9 min
• Building a differential equation from a description ~10 min
• Choosing a model for a scenario ~9 min

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• Second-order linear equations, standard form ~9 min
• The linear differential operator L ~9 min
• Homogeneous versus nonhomogeneous ~8 min
• Linear independence of two functions ~9 min
• The Wronskian ~10 min
• Abel's formula ~9 min
• The superposition principle ~9 min
• Fundamental sets of solutions ~9 min
• The general solution of a linear homogeneous equation ~9 min
• Reduction of order ~11 min

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• The characteristic equation ~9 min
• Distinct real roots ~9 min
• Repeated real roots ~9 min
• Complex roots and Euler's formula ~10 min
• Oscillatory solutions, amplitude-phase form ~9 min
• Exponential growth and decay from real roots ~8 min
• The Cauchy-Euler equation ~10 min
• Cauchy-Euler with repeated and complex roots ~10 min
• Behavior as x goes to infinity ~9 min
• Connecting roots to the shape of a solution ~9 min

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• Mass-spring systems and Hooke's law ~8 min
• Setting up the equation of motion ~8 min
• Undamped free vibration, natural frequency ~8 min
• Damped free vibration ~8 min
• Overdamping ~8 min
• Critical damping ~8 min
• Underdamping and the damping ratio ~9 min
• Energy in a vibrating system ~8 min
• RLC circuits ~8 min
• The mechanical-electrical analogy ~8 min

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• Complementary versus particular solutions ~8 min
• The method of undetermined coefficients ~8 min
• Trial functions for each forcing type ~8 min
• The resonance modification rule ~8 min
• Undetermined coefficients with superposed forcing ~8 min
• Variation of parameters, motivated ~8 min
• Deriving the variation-of-parameters formula ~9 min
• Comparing the two methods ~8 min
• Sinusoidal forcing, steady-state versus transient ~8 min
• Amplitude response and resonance ~8 min
• Phase shift ~8 min
• Beats ~8 min

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• nth-order linear equations ~8 min
• The characteristic polynomial and its roots ~9 min
• Repeated and complex roots at higher order ~9 min
• Superposition and the general solution ~8 min
• Undetermined coefficients at higher order ~9 min
• Variation of parameters at higher order ~9 min
• Converting an nth-order equation to a first-order system ~8 min
• Why systems are the more general object ~7 min

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• Definition and motivation ~9 min
• Existence of the transform ~8 min
• Linearity ~7 min
• Basic transform pairs ~8 min
• The inverse Laplace transform ~8 min
• Transforms of derivatives ~9 min
• s-shifting (exponential shifting) ~8 min
• t-shifting and the unit step function ~9 min
• Piecewise-defined forcing ~9 min
• The Dirac delta and impulse forcing ~9 min
• Convolution and the convolution theorem ~10 min
• Solving an IVP by Laplace transform, start to finish ~10 min

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• Writing a system in matrix form ~8 min
• From a higher-order equation to a system ~8 min
• Vector-valued solutions ~8 min
• The eigenvalue method: distinct real eigenvalues ~10 min
• Complex eigenvalues ~10 min
• Repeated eigenvalues and generalized eigenvectors ~10 min
• Fundamental matrices ~8 min
• The matrix exponential ~9 min
• Solving a system via the matrix exponential ~8 min
• Diagonalization and decoupled systems ~9 min
• Nonhomogeneous linear systems ~9 min

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• Trajectories vs. solution curves ~8 min
• Equilibrium points of a system ~8 min
• Nullclines ~8 min
• Direction fields for systems ~8 min
• Nodes ~9 min
• Saddles ~8 min
• Centers ~8 min
• Spirals ~9 min
• Degenerate and repeated-eigenvalue cases ~9 min
• The trace-determinant plane ~10 min
• Reading stability from trace and determinant alone ~8 min

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• Nonlinear autonomous systems ~9 min
• The Jacobian and local linearization ~10 min
• When linearization fails: Hartman-Grobman ~11 min
• Predator-prey models ~11 min
• Competing species models ~11 min
• The SIR epidemic model ~11 min
• Conserved quantities ~10 min
• Limit cycles ~9 min
• The Van der Pol oscillator ~10 min
• Bifurcations in a system ~10 min
• Choosing a dynamical model for a scenario ~10 min

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• IVPs vs. BVPs, revisited ~10 min
• Two-point boundary conditions ~9 min
• Existence and uniqueness for BVPs, informally ~10 min
• The shooting method ~14 min
• Eigenvalue problems from boundary conditions ~12 min
• Sturm-Liouville form ~12 min
• Orthogonality of eigenfunctions ~13 min
• Expanding a function in a Sturm-Liouville eigenbasis ~14 min
• Separation of variables: a first look toward PDEs ~13 min
• Superposition and the general series solution ~11 min
• Computing the sine-series coefficients for a separated heat equation solution ~15 min

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• Power series review ~10 min
• Ordinary points and the power-series method ~12 min
• Recurrence relations ~12 min
• Radius of convergence of a series solution ~10 min
• Regular singular points ~12 min
• The indicial equation ~12 min
• The Frobenius method ~15 min
• Repeated and differing roots of the indicial equation ~15 min
• Bessel's equation ~14 min
• Legendre's and Hermite's equations ~14 min