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Ordinary Differential Equations
Explore first-order equations; linear equations and forcing; systems and phase space. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Derivatives
A derivative is the limit of a difference quotient, measuring local rate of change when that limit exists.
7 reviewed resources
First-order equations
Ordinary differential equations
An ODE relates an unknown function of one variable to its derivatives.
Video explanations to be addedInitial value problems
Initial data select solutions of a differential equation when existence and uniqueness conditions hold.
Video explanations to be addedSeparable equations
Separation places dependent and independent variables on opposite sides before integration.
Video explanations to be addedIntegrating factors
An integrating factor turns a linear first-order ODE into a product derivative.
Video explanations to be addedExistence and uniqueness for ODEs
Continuity and suitable local Lipschitz conditions ensure a locally unique initial-value solution.
Video explanations to be addedExact differential equations
A differential expression is exact when it is the differential of a potential function on the domain.
Video explanations to be addedBernoulli equations
A power substitution converts a suitable nonlinear first-order Bernoulli equation into a linear one.
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Linear equations and forcing
Second-order linear ODEs
A second-order linear ODE combines a function and its first two derivatives linearly.
Video explanations to be addedCharacteristic equations
An exponential trial converts a constant-coefficient homogeneous linear ODE into a polynomial equation.
Video explanations to be addedVariation of parameters
Variable coefficients multiplying homogeneous solutions construct a particular solution of a forced linear ODE.
Video explanations to be addedLaplace transform
An integral transform converts time-domain derivatives into algebraic relations under convergence conditions.
Video explanations to be addedResonance
Forcing near a natural frequency can produce large oscillatory responses depending on damping.
Video explanations to be addedWronskian
A Wronskian tests independence of solutions under differential-equation hypotheses; arbitrary functions require care.
Video explanations to be addedUndetermined coefficients
A forcing-compatible trial form finds particular solutions of suitable constant-coefficient linear equations.
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Systems and phase space
Systems of ODEs
Coupled differential equations describe interacting state variables.
2 reviewed resourcesPhase portraits
A phase portrait displays trajectories in state space rather than against time.
Video explanations to be addedEquilibria and stability
Stability describes whether nearby trajectories remain close to or approach an equilibrium.
Video explanations to be addedLinearization
The Jacobian approximates a smooth system near an equilibrium; zero-real-part eigenvalues need further analysis.
1 reviewed resourcesEigenvalues and eigenvectors
A nonzero eigenvector is scaled by a linear operator; the scale factor is its eigenvalue.
3 reviewed resources