Enhance Your Learning with Differential Equations Flash Cards for quick revision
An equation that relates a function with its derivatives, representing a mathematical model for various physical, biological, and social phenomena.
A differential equation that involves only the first derivative of the unknown function.
A differential equation that involves the second derivative of the unknown function.
A differential equation that involves derivatives of order higher than two.
A differential equation that can be expressed in the form of a linear combination of the unknown function and its derivatives.
A differential equation that cannot be expressed in the form of a linear combination of the unknown function and its derivatives.
A differential equation in which all terms can be expressed as a linear combination of the unknown function and its derivatives.
A differential equation in which at least one term cannot be expressed as a linear combination of the unknown function and its derivatives.
A differential equation with additional conditions specified at the boundaries of the domain.
A differential equation with additional conditions specified at a single point in the domain.
The process of finding the unknown function that satisfies a given differential equation.
The use of differential equations to model and solve real-world problems in various fields such as physics, engineering, biology, and economics.
Techniques for approximating the solutions of differential equations using numerical algorithms.
A differential equation that involves partial derivatives of the unknown function with respect to multiple independent variables.
A set of differential equations that are interconnected and need to be solved simultaneously.
The use of Fourier series to solve certain types of differential equations with periodic boundary conditions.
The use of Laplace transform to solve differential equations by transforming them into algebraic equations.
The use of power series expansions to find solutions of differential equations as infinite series.
The study of the stability and behavior of solutions of differential equations using phase plane analysis.
The conditions under which a differential equation has a unique solution or multiple solutions.
A numerical method for approximating the solutions of first-order differential equations using iterative calculations.
A family of numerical methods for solving ordinary differential equations by approximating the solution at multiple points within each step.
A technique used to solve certain types of first-order ordinary differential equations by separating the variables on different sides of the equation.
An equation obtained by substituting a trial solution into a linear homogeneous differential equation, used to find the roots and form the general solution.
The values and corresponding vectors that satisfy a certain equation, used to solve systems of linear differential equations.
A graphical representation of the behavior and trajectories of solutions of a system of differential equations in the phase plane.
A point in the phase plane where the system of differential equations has a constant solution.
A condition in which the system of differential equations remains unchanged over time.
A solution of a differential equation that repeats itself after a certain period of time.
Oscillations that decrease in amplitude over time due to the presence of damping in the system.
Oscillations that are influenced by an external force or input in addition to the natural behavior of the system.
A phenomenon in which the amplitude of oscillations becomes significantly larger when the frequency of the external force matches the natural frequency of the system.
A mathematical theory used to study the properties of second-order linear homogeneous differential equations with boundary conditions.
A function used to solve inhomogeneous linear differential equations with specified boundary conditions.
A partial differential equation that describes the distribution of heat in a given region over time.
A partial differential equation that describes the propagation of waves in a given region over time.
A partial differential equation that arises in various physical and mathematical contexts, including electrostatics and fluid dynamics.
Functions that satisfy Laplace's equation, often used to model physical phenomena such as temperature distribution and fluid flow.
A set of functions that are mutually perpendicular or independent with respect to a given inner product.
A family of solutions to Bessel's differential equation, often used to solve problems involving cylindrical symmetry.
A family of orthogonal polynomials that arise in the solution of Laplace's equation in spherical coordinates.
A family of orthogonal polynomials that arise in the approximation of functions and the solution of differential equations.
A family of orthogonal polynomials that arise in the solution of quantum harmonic oscillator and heat conduction problems.
A family of orthogonal polynomials that arise in the solution of the hydrogen atom and other quantum mechanical problems.
A second-order linear differential equation that has solutions in terms of hypergeometric functions.
A second-order linear differential equation that has solutions in terms of Airy functions, often used to describe oscillatory phenomena.
A second-order linear differential equation that has solutions in terms of Bessel functions, often used to describe wave propagation and diffraction.
A second-order linear differential equation that has solutions in terms of Legendre polynomials, often used to describe physical phenomena with spherical symmetry.
A first-order nonlinear ordinary differential equation that can be transformed into a linear second-order equation through a suitable change of variables.
A first-order nonlinear ordinary differential equation that can be transformed into a linear equation through a suitable change of variables.
A differential equation that can be expressed as the total differential of a function, allowing for an exact solution.
A technique used to reduce the order of a linear homogeneous differential equation by finding a second linearly independent solution.
A method used to find a particular solution of a nonhomogeneous linear differential equation by assuming a solution of the form y = u1(x)y1(x) + u2(x)y2(x).
A determinant used to determine the linear independence of a set of solutions of a linear homogeneous differential equation.
A second-order linear homogeneous differential equation with variable coefficients, often used to solve problems in physics and engineering.
A second-order linear homogeneous differential equation with variable coefficients, often used to solve problems in physics and engineering.
A method used to find power series solutions of second-order linear differential equations with variable coefficients, including equations with regular singular points.
A second-order linear differential equation that arises in problems involving cylindrical symmetry, often used to describe wave propagation and diffraction.
A second-order linear differential equation that arises in problems with spherical symmetry, often used to describe physical phenomena.
A special function that arises in the solution of linear differential equations, often used to represent solutions in terms of power series.
A special function that arises in the solution of linear differential equations, often used to describe oscillatory phenomena.
A special function that extends the concept of factorial to real and complex numbers, often used in the solution of differential equations.
A family of special functions that arise in the solution of linear differential equations, often used to describe wave propagation and diffraction.
A family of orthogonal polynomials that arise in the solution of linear differential equations, often used to describe physical phenomena with spherical symmetry.
A family of orthogonal polynomials that arise in the solution of linear differential equations, often used in approximation theory and numerical analysis.
A family of orthogonal polynomials that arise in the solution of linear differential equations, often used to describe quantum harmonic oscillators and heat conduction.
A family of orthogonal polynomials that arise in the solution of linear differential equations, often used to describe the hydrogen atom and other quantum mechanical systems.
An infinite series that arises in the solution of linear differential equations, often used to represent solutions in terms of power series.
A solution of a differential equation that can be expressed as an infinite series, often used to find approximate solutions.
A solution of a differential equation that can be expressed as a series, often used to find approximate solutions.