Ordinary Differential Equations Vocabulary

Vocabulary terms and definitions from the lecture notes on Ordinary Differential Equations, covering first and second order equations, Laplace transforms, and linear systems.

Differential Equation

An equation where the unknown is a function and both the function and its derivatives may appear in the equation.

Order

The highest derivative order that appears in a differential equation.

Ordinary Differential Equation (ODE)

A differential equation where the unknown function depends on a single independent variable, usually denote as $t$.

Partial Differential Equation (PDE)

A differential equation where the unknown function depends on two or more independent variables, and their partial derivatives appear in the equation.

Linear Differential Equation

An ODE is linear if the source function $f$ is linear on its second argument, resulting in the form $y'(t) = a(t)y + b(t)$. Otherwise, it has variable coefficients or is nonlinear.

Initial Value Problem (IVP)

The problem of finding a solution to a differential equation that satisfies specific conditions at a given point, such as $y(t_0) = y_0$ and potentially its derivatives.

Integrating Factor Method

A method used to solve linear first-order differential equations by multiplying the equation by a function $\mu(t)$ to transform one side into a total derivative.

Potential Function (\psi)

A function whose total derivative with respect to the independent variable is zero, converting a differential equation into the form $\psi' = 0$.

Bernoulli Equation

A first-order non-linear differential equation of the form $y' = p(t)y + q(t)y^n$, which can be transformed into a linear equation.

Separable Differential Equation

A differential equation that can be written in the form $h(y)y' = g(t)$, where the left side depends only on $y$ and the right side only on $t$.

Euler Homogeneous Equation

A first-order differential equation of the form $y'(t) = F(y(t)/t)$, often solved by transforming it into a separable equation.

Exact Differential Equation

A differential equation $N(t, y)y' + M(t, y) = 0$ where the partial derivatives satisfy $\partial_t N(t, y) = \partial_y M(t, y)$, and which represents a total derivative of a potential function.

Half-life (\tau)

The time it takes for half of a radioactive material to decay, related to the decay constant $k$ by the formula $k\tau = \ln(2)$.

Newton's Cooling Law

The principle stating that the rate of temperature change of an object is proportional to the difference between its temperature and the constant temperature of the surrounding medium.

Linear Operator (L)

A function acting on other functions that satisfies the property $L(c_1y_1 + c_2y_2) = c_1L(y_1) + c_2L(y_2)$ for arbitrary constants and functions.

Superposition Property

A property of linear homogeneous equations where a linear combination of any two solutions is also a solution.

Linearly Dependent Functions

Two functions $y_1, y_2$ are linearly dependent if they are proportional to each other, such that $y_1(t) = c y_2(t)$ for all $t$.

Fundamental Solutions

A set of functions $y_1, y_2$ that are linearly independent solutions of an homogeneous linear differential equation.

Wronskian

The determinant of the matrix formed by a set of functions and their derivatives, given as $W_{12}(t) = y_1(t)y'_2(t) - y'_1(t)y_2(t)$.

Abel's Theorem

A theorem stating that the Wronskian of two solutions to a linear second-order homogeneous ODE satisfies the first-order linear equation $W' + a_1(t)W = 0$.

Characteristic Polynomial

For a second-order linear homogeneous ODE with constant coefficients $y'' + a_1y' + a_0y = 0$, it is the polynomial $p(r) = r^2 + a_1r + a_0$.

Euler Equidimensional Equation

A linear second-order ODE of the form $(t - t_0)^2 y'' + a_1(t - t_0)y' + a_0y = 0$ where the power of the independent variable matches the order of the derivative.

Variation of Parameters Method

A general method to find a particular solution $y_p$ to a nonhomogeneous linear ODE using the fundamental solutions of the corresponding homogeneous equation.

Undetermined Coefficients Method

A method to find a particular solution $y_p$ of a nonhomogeneous constant-coefficient linear ODE by guessing a form based on the source function $f(t)$.

Regular Point

A point $x_0$ where the coefficient functions $p(x)$ and $q(x)$ of a linear second-order differential equation are analytic.

Regular Singular Point

A singular point $x_0$ of a linear second-order ODE where the functions $(x - x_0)p(x)$ and $(x - x_0)^2q(x)$ are analytic.

Laplace Transform

An integral transformation of a function $f(t)$ defined as $F(s) = \int_0^\infty e^{-st}f(t) dt$, useful for converting constant-coefficient ODEs into algebraic equations.

Step Function (u)

A discontinuous function defined as $u(t) = 0$ for $t < 0$ and $u(t) = 1$ for $t \ge 0$, used to construct piecewise continuous sources.

Dirac Delta Generalized Function (\delta)

A generalized function defined as the limit of a sequence of bump functions, characterized by the property $\int_a^b f(t) \delta(t - c) dt = f(c)$.

Convolution

A binary operation on two functions defined as $(f * g)(t) = \int_0^t f(\tau)g(t - \tau) d\tau$.

Fundamental Matrix

An $n \times n$ matrix valued function $X(t)$ whose columns are formed by a fundamental set of solutions to a linear differential system.

Critical Point

A point $y_c$ in an autonomous system $y' = f(y)$ such that $f(y_c) = 0$, corresponding to stationary or equilibrium solutions.

Jacobian Matrix

The matrix of first-order partial derivatives of a vector field, used in the linearization of nonlinear systems at critical points.

Hyperbolic Critical Point

A critical point of a two-dimensional nonlinear system where the linearized Jacobian matrix has eigenvalues with non-zero real parts.

Diagonalizable Matrix

A square matrix $A$ that can be decomposed as $A = PDP^{-1}$ where $D$ is a diagonal matrix and $P$ is invertible.

Matrix Exponential ($e^A$)

An infinite sum defined as $e^A = \sum_{n=0}^\infty \frac{A^n}{n!}$, used to express solutions of first-order linear homogeneous systems with constant coefficients.