ODEs Definitions

Question 1

What is a differential equation?

Answer 1

An equation that relates a function with one or more of its derivatives.

Question 2

What is the order of a differential equation?

Answer 2

The order of a differential equation is the highest order derivative present in the equation.

Question 3

True or False: A differential equation can involve multiple variables.

Answer 3

True

Question 4

What is the general form of a first-order differential equation?

Answer 4

dy/dx = f(x, y)

Question 5

What is an ordinary differential equation?

Answer 5

An equation involving derivatives of a function with respect to a single independent variable.

Question 6

What is a partial differential equation?

Answer 6

An equation involving partial derivatives of a function with respect to multiple independent variables.

Question 7

What is the solution to a differential equation?

Answer 7

A function or a set of functions that satisfy the equation.

Question 8

What is an initial condition in the context of differential equations?

Answer 8

A condition that specifies the value of the function at a particular point.

Question 9

What is a boundary condition in the context of differential equations?

Answer 9

A condition that specifies the value of the function at the boundaries of the domain.

Question 10

What is an exact differential equation?

Answer 10

An equation that can be expressed as the differential of a function.

Question 11

What is an implicit differential equation?

Answer 11

An equation where the dependent and independent variables are not explicitly separated.

Question 12

What is a linear differential equation?

Answer 12

An equation that is linear in the dependent variable and its derivatives.

Question 13

What is a homogeneous differential equation?

Answer 13

An equation where all terms involve the dependent variable and its derivatives.

Question 14

What is a non-linear differential equation?

Answer 14

An equation that is not linear in the dependent variable and its derivatives.

Question 15

What is the characteristic equation of a linear differential equation?

Answer 15

An equation obtained by substituting y = e^(mx) into the differential equation.

Question 16

What is the Wronskian of two functions in the context of differential equations?

Answer 16

A determinant used to determine linear independence of the functions.

Question 17

What is the method of undetermined coefficients?

Answer 17

A method used to find a particular solution to a non-homogeneous linear differential equation.

Question 18

What is the method of variation of parameters?

Answer 18

A method used to find a particular solution to a non-homogeneous linear differential equation by assuming a solution of a specific form.

Question 19

What is the Laplace transform in the context of differential equations?

Answer 19

A method used to transform a differential equation into an algebraic equation.

Question 20

What is the Fourier transform in the context of differential equations?

Answer 20

A method used to transform a function of time into a function of frequency.

Question 21

What is the method of characteristics used for in solving differential equations?

Answer 21

A method used for first-order partial differential equations to reduce them to a system of ordinary differential equations.

Question 22

What is a singular solution in the context of differential equations?

Answer 22

A solution that does not satisfy the general solution of the differential equation.

Question 23

What is the method of separation of variables?

Answer 23

A method used to solve first-order ordinary differential equations by separating variables on opposite sides of the equation.

Question 24

What is the Picard-Lindelöf theorem?

Answer 24

A theorem that guarantees the existence and uniqueness of solutions to certain initial value problems for ordinary differential equations.

Question 25

What is an autonomous differential equation?

Answer 25

An equation where the independent variable does not explicitly appear.

Question 26

What is the concept of stability in the context of differential equations?

Answer 26

The property of a solution that remains bounded as time approaches infinity.

Question 27

What is the method of power series used for in solving differential equations?

Answer 27

A method used to find a series solution for differential equations around a singular point.

Question 28

What is the Green’s function in the context of differential equations?

Answer 28

A function used to solve inhomogeneous differential equations with specific boundary conditions.