Glossary of Key Terms
Glossary
This page collects the key terms, concepts, and formulas used throughout the course in one place, organized alphabetically. Each entry gives a short, informal description along with a link to the Notes or Additional Topics section where the idea is developed more fully. Use this page as a lookup tool, not as a substitute for reading the notes.
Looking for where a term first shows up, or need a quick reminder of what a formula means? Search this page (Ctrl+F / Cmd+F) for the term, then follow the link to the full treatment in the Notes or Additional Topics.
A
Amplitude
In an oscillating solution such as \(x(t) = A\cos(\omega t - \phi)\), the amplitude \(A\) is the maximum displacement from equilibrium. Shows up when writing solutions of the spring-mass and RLC models in amplitude-phase form.
Asymptotically Stable Equilibrium
An equilibrium solution that solutions nearby not only stay near, but actually approach as \(t \to \infty\). On a phase line or phase plane, nearby trajectories are drawn into the equilibrium.
Autonomous Equation
A differential equation in which the independent variable (usually \(t\)) does not appear explicitly — only the dependent variable and its derivatives do, e.g. \(y' = f(y)\). Autonomous equations are the setting for phase-line and equilibrium analysis.
→ See: Notes 3
B
Bernoulli Equation
A first-order nonlinear equation of the form \(y' + p(t)y = q(t)y^n\) that can be converted into a linear equation via the substitution \(v = y^{1-n}\).
→ See: Additional Topics 2
Bifurcation
A qualitative change in the number or stability of equilibrium points as a parameter in the differential equation is varied. Often visualized with a bifurcation diagram plotting equilibria against the parameter.
→ See: Notes 3
C
Carrying Capacity
In the logistic growth model \(P' = rP\left(1 - \frac{P}{K}\right)\), the constant \(K\) that a population approaches as \(t \to \infty\) — the maximum sustainable population size.
→ See: Notes 3
Characteristic Equation
For a linear constant-coefficient ODE (or system), the algebraic equation obtained by substituting a trial exponential solution. For \(ay'' + by' + cy = 0\) this is \(ar^2 + br + c = 0\); for a linear system \(\mathbf{x}' = A\mathbf{x}\) it is \(\det(A - \lambda I) = 0\). Its roots determine the form of the general solution.
Continuous Dependence
The (informal) idea that small changes in the initial condition of an IVP produce only small changes in the solution, at least over finite time intervals. Part of why numerical and modeling predictions are meaningful in practice.
→ See: Additional Topics 4
Convolution
An operation combining two functions, \((f * g)(t) = \int_0^t f(\tau)g(t-\tau)\,d\tau\), that corresponds under the Laplace transform to ordinary multiplication: \(\mathcal{L}\{f * g\} = F(s)G(s)\). Used to solve for the response of a system to an arbitrary forcing function once its response to an impulse is known.
→ See: Notes 8
Critically Damped
A second-order linear system (e.g., spring-mass-damper) for which the characteristic equation has a repeated real root. The system returns to equilibrium as fast as possible without oscillating.
→ See: Notes 4, Additional Topics 5
D
Damping
A dissipative effect (e.g., friction, resistance) in a physical system that removes energy over time. In the spring-mass equation \(mx'' + \gamma x' + kx = 0\), \(\gamma\) is the damping coefficient; its size relative to \(m\) and \(k\) determines whether the system is overdamped, underdamped, or critically damped.
→ See: Notes 4, Additional Topics 5
Determinant
A scalar computed from a square matrix that, among other things, tells you whether the matrix is invertible (\(\det A \neq 0\)) and appears directly in the characteristic equation of a linear system.
→ See: Notes 10, Additional Topics 8
Differential Equation
An equation relating a function to one or more of its derivatives. The “unknown” you solve for is not a number but a function.
→ See: Notes 1
Dimensional Analysis
A technique for using the physical units of the quantities in a model to check consistency, reduce the number of parameters, and rewrite an equation in dimensionless form.
→ See: Additional Topics 1
Direction Field (Slope Field)
A graphical representation of a first-order ODE \(y' = f(t,y)\) obtained by drawing a short line segment of slope \(f(t,y)\) at many points in the \(ty\)-plane. Solution curves are curves that stay tangent to these segments everywhere.
→ See: Notes 2
E
Eigenvalue / Eigenvector
For a square matrix \(A\), a scalar \(\lambda\) and nonzero vector \(\mathbf{v}\) satisfying \(A\mathbf{v} = \lambda \mathbf{v}\). For a linear system \(\mathbf{x}' = A\mathbf{x}\), eigenvalues determine the growth/decay/oscillation rates of solutions and eigenvectors determine the directions of straight-line solutions.
→ See: Notes 10, Additional Topics 8
Equilibrium Point (Equilibrium Solution)
A constant solution of a differential equation or system — a point where the rate of change is zero. Found by setting the right-hand side of the equation(s) equal to zero and solving.
Euler’s Method
The simplest numerical method for approximating the solution of an IVP: starting from \((t_0, y_0)\), step forward using \(y_{n+1} = y_n + h\,f(t_n, y_n)\), where \(h\) is the step size. Geometrically, it follows the direction field in straight-line steps.
→ See: Notes 14
Existence and Uniqueness
Informally, conditions (typically continuity and smoothness of \(f\)) under which an IVP \(y' = f(t,y)\), \(y(t_0) = y_0\) is guaranteed to have exactly one solution, at least locally in time.
→ See: Additional Topics 4
F
Fundamental Set of Solutions
For an \(n\)th-order linear homogeneous ODE, a collection of \(n\) linearly independent solutions whose linear combinations give every solution of the equation (the general solution).
→ See: Notes 4
G
General Solution
A solution to a differential equation that contains arbitrary constants (one for each order of the equation) and represents the entire family of solutions. Fixing the constants using initial or boundary conditions gives a particular solution.
H
Heaviside (Unit Step) Function
The function \(u(t-a)\) that is \(0\) for \(t < a\) and \(1\) for \(t \geq a\). Used to build piecewise/on-off forcing terms for use with Laplace transforms.
→ See: Notes 8
Homogeneous Equation
A linear differential equation in which every term involves the unknown function or its derivatives — equivalently, the right-hand side is zero (e.g., \(y'' + p(t)y' + q(t)y = 0\)). Contrast with a nonhomogeneous equation, which has a nonzero forcing term.
Hooke’s Law
The linear restoring-force law for an idealized spring, \(F = -kx\), where \(k > 0\) is the spring constant. The starting point for deriving the spring-mass differential equation.
→ See: Additional Topics 5
I
Impulse Function (Dirac Delta)
An idealized “instantaneous kick,” denoted \(\delta(t-a)\), used to model forces or inputs that act over a vanishingly short time but deliver a finite total effect. Its Laplace transform is simply \(e^{-as}\).
→ See: Notes 8
Initial Condition
A specified value of the unknown function (and possibly its derivatives) at a starting point, e.g. \(y(t_0) = y_0\). Used together with a differential equation to pick out a single particular solution from the general solution.
→ See: Notes 1
Initial Value Problem (IVP)
A differential equation paired with initial condition(s) sufficient to determine a unique solution.
→ See: Notes 1
Integrating Factor
A function \(\mu(t)\) used to convert a linear first-order ODE \(y' + p(t)y = q(t)\) into a form where the left-hand side is an exact derivative, \(\frac{d}{dt}[\mu(t)y]\). Given by \(\mu(t) = e^{\int p(t)\,dt}\).
→ See: Notes 2
J
Jacobian Matrix
For a nonlinear system \(\mathbf{x}' = \mathbf{f}(\mathbf{x})\), the matrix of first partial derivatives of \(\mathbf{f}\). Evaluated at an equilibrium point, it gives the linear system that approximates the nonlinear dynamics near that point.
K
Kirchhoff’s Voltage Law
The circuit principle that the sum of voltage drops around any closed loop equals zero. Used to derive the differential equation governing an RLC circuit.
→ See: Additional Topics 6
L
Laplace Transform
An integral transform, \(\mathcal{L}\{f(t)\} = F(s) = \int_0^\infty e^{-st}f(t)\,dt\), that converts a linear ODE with constant coefficients into an algebraic equation in \(s\). Solving the algebraic equation and transforming back (inverse Laplace transform) gives the solution to the original ODE.
Linear Combination
A sum of scalar multiples of functions or vectors, e.g. \(c_1 y_1(t) + c_2 y_2(t)\). The general solution of a linear homogeneous equation is a linear combination of a fundamental set of solutions.
→ See: Notes 4, Additional Topics 3
Linear Equation / Linear System
A differential equation (or system) in which the unknown function(s) and their derivatives appear only to the first power and are not multiplied together. Linear equations obey the superposition principle, which is what makes them especially tractable.
→ See: Notes 2, Notes 9, Additional Topics 3
Linearization
The process of approximating a nonlinear system near an equilibrium point by the linear system given by its Jacobian matrix evaluated at that point. Valid (and useful for classifying stability) as long as the equilibrium is not a borderline case.
Linear Independence
A set of functions or vectors is linearly independent if no one of them can be written as a linear combination of the others. Needed for a set of solutions to qualify as a fundamental set.
→ See: Notes 4, Additional Topics 8
M
Matrix Exponential
Not covered in depth in this course, but worth knowing by name: the generalization \(e^{At}\) of the scalar exponential to matrices, giving a compact way to write the general solution \(\mathbf{x}(t) = e^{At}\mathbf{x}_0\) of a linear system.
→ See: Notes 10 (context)
N
Node
A type of equilibrium point in a planar linear (or linearized) system in which all nearby trajectories approach (stable node) or move away from (unstable node) the equilibrium without spiraling, corresponding to real eigenvalues of the same sign.
Nonlinear Equation / System
A differential equation or system that is not linear — typically because the unknown function appears in a product, power, or inside another function (e.g., \(\sin y\), \(y^2\), \(y_1 y_2\)). Nonlinear systems generally can’t be solved in closed form and are studied qualitatively.
Nullcline
For a planar system \(x' = f(x,y)\), \(y' = g(x,y)\), the curve where \(f(x,y) = 0\) (the \(x\)-nullcline) or \(g(x,y) = 0\) (the \(y\)-nullcline). Equilibria occur where an \(x\)-nullcline and \(y\)-nullcline intersect, and nullclines divide the plane into regions of consistent flow direction.
→ See: Notes 12
O
Order (of a Differential Equation)
The order of the highest derivative appearing in the equation. For example, \(y'' + 3y' = \sin t\) is second order.
→ See: Notes 1
Overdamped
A second-order linear system whose characteristic equation has two distinct real roots. The system returns to equilibrium without oscillating, more slowly than the critically damped case.
→ See: Notes 4, Additional Topics 5
P
Particular Solution
A single specific solution to a differential equation, either (a) one obtained by fixing the arbitrary constants in a general solution using initial conditions, or (b) in the context of nonhomogeneous equations, any one solution that satisfies the full nonhomogeneous equation (to be added to the homogeneous general solution).
Phase Line
A one-dimensional diagram for an autonomous first-order equation \(y' = f(y)\), showing equilibrium points on a line along with arrows indicating whether solutions increase or decrease in the intervals between them. The basic tool for qualitative analysis of first-order autonomous equations.
→ See: Notes 3
Phase Plane / Phase Portrait
The plane whose axes are the state variables of a system (e.g., \(x\) and \(y\)), used to plot trajectories of solutions over time without explicit reference to \(t\). A phase portrait is a representative collection of such trajectories showing the overall qualitative behavior of the system.
Power Series Solution
A method for solving a linear ODE by assuming a solution of the form \(y(t) = \sum_{n=0}^\infty a_n (t - t_0)^n\) and solving for the coefficients via a recurrence relation. Useful when an equation has no closed-form solution in terms of elementary functions.
→ See: Additional Topics 7
R
Resonance
The phenomenon in a forced oscillating system where the forcing frequency matches the system’s natural frequency, causing the amplitude of oscillation to grow (in the undamped case, without bound).
→ See: Notes 5
Runge-Kutta Method (RK4)
A numerical method for approximating IVP solutions that is significantly more accurate than Euler’s method for the same step size, by sampling the slope \(f(t,y)\) at several points within each step and combining them in a weighted average.
→ See: Notes 14
S
Saddle Point
A type of equilibrium point where solutions approach along one direction but move away along another, corresponding to real eigenvalues of opposite sign in a linear (or linearized) system. Always unstable.
Separable Equation
A first-order ODE that can be written in the form \(\frac{dy}{dx} = f(x)g(y)\), allowing the variables to be separated and each side integrated independently.
→ See: Notes 2
Spiral (Focus)
A type of equilibrium point where trajectories wind around the equilibrium while approaching it (stable spiral) or moving away from it (unstable spiral), corresponding to complex eigenvalues with nonzero real part.
Stability
A qualitative description of how solutions behave near an equilibrium point: whether nearby solutions stay close (stable), are drawn in (asymptotically stable), or move away (unstable) as \(t \to \infty\).
Straight-Line Solution
For a linear system \(\mathbf{x}' = A\mathbf{x}\), a solution of the form \(\mathbf{x}(t) = e^{\lambda t}\mathbf{v}\), where \(\lambda\) is an eigenvalue and \(\mathbf{v}\) the corresponding eigenvector. Its trajectory in the phase plane lies along a straight line through the origin.
Superposition Principle
For linear homogeneous equations, the fact that if \(y_1\) and \(y_2\) are solutions, then so is any linear combination \(c_1y_1 + c_2y_2\). The foundation for building general solutions out of fundamental sets.
→ See: Additional Topics 3
System of Differential Equations
A collection of two or more differential equations involving two or more unknown functions, coupled together. Often written in vector form \(\mathbf{x}' = \mathbf{f}(\mathbf{x}, t)\).
→ See: Notes 9
T
Transfer Function
In the Laplace-transform picture of a linear system, the ratio \(H(s) = Y(s)/X(s)\) of the transformed output to the transformed input, encoding how the system responds to any input signal.
→ See: Notes 6, Additional Topics 6
Trajectory
The path traced out in the phase plane (or phase space) by a solution \(\mathbf{x}(t)\) as \(t\) varies. Distinguished from the solution itself, which also carries information about the rate at which the path is traced.
U
Underdamped
A second-order linear system whose characteristic equation has complex conjugate roots. The system oscillates with a slowly (exponentially) decaying amplitude while returning toward equilibrium.
→ See: Notes 4, Additional Topics 5
Unstable Equilibrium
An equilibrium point such that solutions starting arbitrarily close to it move away as \(t \to \infty\), at least for some nearby initial conditions.
V
Variation of Parameters
A method for finding a particular solution to a nonhomogeneous linear ODE that works even when the method of undetermined coefficients doesn’t apply (e.g., non-constant coefficients or non-polynomial/exponential/trig forcing). Builds the particular solution out of the homogeneous solutions with variable (rather than constant) coefficients.
→ See: Notes 5
W
Wronskian
For a set of functions \(y_1, \dots, y_n\), the determinant of the matrix formed from the functions and their derivatives up to order \(n-1\). A nonzero Wronskian at some point confirms that the functions are linearly independent, and hence form a fundamental set of solutions.
→ See: Notes 4
Quick Index by Course Section
For students who prefer to browse by where a term first appears rather than alphabetically:
| Section | Key Terms Introduced |
|---|---|
| Notes 1 | differential equation, order, general/particular solution, IVP, initial condition |
| Notes 2 | direction field, separable equation, linear first-order equation, integrating factor |
| Notes 3 | autonomous equation, equilibrium point, stability, phase line, bifurcation, carrying capacity |
| Notes 4 | characteristic equation, homogeneous equation, fundamental set of solutions, Wronskian, damping (over/under/critical), resonance* |
| Notes 5 | variation of parameters, resonance |
| Notes 6–7 | Laplace transform, Heaviside function |
| Notes 8 | convolution, impulse function (Dirac delta) |
| Notes 9 | system of differential equations, linear system, straight-line solution |
| Notes 10 | eigenvalue, eigenvector, characteristic equation (systems) |
| Notes 11 | phase plane, trajectory, node, saddle point, spiral |
| Notes 12 | nullcline, nonlinear system, linearization, Jacobian matrix |
| Notes 13 | linearization (applications) |
| Notes 14 | Euler’s method, Runge-Kutta method |
| Additional Topics 1 | dimensional analysis |
| Additional Topics 2 | Bernoulli equation |
| Additional Topics 3 | linear combination, superposition principle |
| Additional Topics 4 | existence and uniqueness, continuous dependence |
| Additional Topics 5 | Hooke’s law, amplitude, damping |
| Additional Topics 6 | Kirchhoff’s voltage law, transfer function |
| Additional Topics 7 | power series solution |
| Additional Topics 8 | determinant, eigenvalue/eigenvector, linear independence |
*: Resonance is introduced in the context of Notes 4/5 together — adjust the table above if your actual section boundaries differ.