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.
