ODEs Definitions Flashcards
What is a differential equation?
An equation that relates a function with one or more of its derivatives.
What is the order of a differential equation?
The order of a differential equation is the highest order derivative present in the equation.
True or False: A differential equation can involve multiple variables.
True
What is the general form of a first-order differential equation?
dy/dx = f(x, y)
What is an ordinary differential equation?
An equation involving derivatives of a function with respect to a single independent variable.
What is a partial differential equation?
An equation involving partial derivatives of a function with respect to multiple independent variables.
What is the solution to a differential equation?
A function or a set of functions that satisfy the equation.
What is an initial condition in the context of differential equations?
A condition that specifies the value of the function at a particular point.
What is a boundary condition in the context of differential equations?
A condition that specifies the value of the function at the boundaries of the domain.
What is an exact differential equation?
An equation that can be expressed as the differential of a function.
What is an implicit differential equation?
An equation where the dependent and independent variables are not explicitly separated.
What is a linear differential equation?
An equation that is linear in the dependent variable and its derivatives.
What is a homogeneous differential equation?
An equation where all terms involve the dependent variable and its derivatives.
What is a non-linear differential equation?
An equation that is not linear in the dependent variable and its derivatives.
What is the characteristic equation of a linear differential equation?
An equation obtained by substituting y = e^(mx) into the differential equation.
What is the Wronskian of two functions in the context of differential equations?
A determinant used to determine linear independence of the functions.
What is the method of undetermined coefficients?
A method used to find a particular solution to a non-homogeneous linear differential equation.
What is the method of variation of parameters?
A method used to find a particular solution to a non-homogeneous linear differential equation by assuming a solution of a specific form.
What is the Laplace transform in the context of differential equations?
A method used to transform a differential equation into an algebraic equation.
What is the Fourier transform in the context of differential equations?
A method used to transform a function of time into a function of frequency.
What is the method of characteristics used for in solving differential equations?
A method used for first-order partial differential equations to reduce them to a system of ordinary differential equations.
What is a singular solution in the context of differential equations?
A solution that does not satisfy the general solution of the differential equation.
What is the method of separation of variables?
A method used to solve first-order ordinary differential equations by separating variables on opposite sides of the equation.
What is the Picard-Lindelöf theorem?
A theorem that guarantees the existence and uniqueness of solutions to certain initial value problems for ordinary differential equations.
What is an autonomous differential equation?
An equation where the independent variable does not explicitly appear.
What is the concept of stability in the context of differential equations?
The property of a solution that remains bounded as time approaches infinity.
What is the method of power series used for in solving differential equations?
A method used to find a series solution for differential equations around a singular point.
What is the Green’s function in the context of differential equations?
A function used to solve inhomogeneous differential equations with specific boundary conditions.
