Definitions For Final

Question 1

What is a Ordinary Differential Equation

Answer 1

A DE that contains only derivatives of one or more unknown functions with respect to only one independent variable (Could be many dependent variables, but only one independent variable)

Question 2

What is a Differential Equation?

Answer 2

An equation with the derivatives of one or more unknown functions (or dependent variables) with respect to one t more independent variables.

Question 3

What is a Partial Differential Equation?

Answer 3

If a differential equation involves partial derivatives of one or more unknown functions of two or more independent variables, it is said to be an partial differential
equation (PDE). (Has derivatives with respect to 2 different independent variables)

Question 4

What is Type of a DE?

Answer 4

Defines whether the DE is Ordinary or Partial

Question 5

What is the Order of a DE?

Answer 5

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

Question 6

What is a Linear Differential Equation?

Answer 6

An nth order ODE is said to be linear if it can be written in linear form.

Question 7

What is a Nonlinear DE?

Answer 7

An ODE is said to be nonlinear if it is not linear.

Question 8

What does it mean for a DE to be in Normal Form?

Answer 8

A DE is in normal form if, for a general DE, it is in the form that all of its derivatives are arranged in ascending order, and all of the function is on one side, so that the DE is of the form LHS = 0. For an ODE, normal form is of the form
highest order derivative = everything else in ascending order.

Question 9

What is a Solution to an ODE?

Answer 9

A function that is n-differentiable on an interval of definition I, that when substituted into the DE results in an identity over I.

Question 10

Trivial Solution

Answer 10

A solution of a differential equation that is identically zero on an interval I is said to be a trivial solution on I ( a solution that satisfies the DE and results in the DE always equaling 0 over the interval I)

Question 11

Constant Solution

Answer 11

A solution of a differential equation that is identically a constant on an interval I is said to be a constant solution on I. (A solution that satisfies the DE and results in the DE always equaling a constant over the interval I)

Question 12

What is an Explicit Solution?

Answer 12

A solution to an ODE in which the dependent variable is expressed solely in terms of the independent variable and constants ( a solution in the form y = …)

Question 13

What is an Implicit Solution?

Answer 13

A relation G(x, y) = 0 is said to be an implicit solution of an ODE on an interval I, provided there exists at least one function ϕ that satisfies the relation as well as
the differential equation on I. (A relation of x and y in the form G(x, y) = 0, that contains at least one function (that satisfies the relation) that is also a solution to the given DE on the interval I)

Question 14

How do you verify a solution to a DE?

Answer 14

To verify an explicit solution y = ϕ(x) to an ODE on an interval I, one should demonstrate that the equation holds for every value of x in I when y and its derivatives
are replaced by ϕ(x) and its derivatives (plug in the function and show that it results in an identity)

To verify an implicit solution you can either do:

1. Find an explicit solution out of the implicit solution and show the same way as for an explicit solution
2. Use implicit differentiation to show that if the implicit relationship holds, the ODE is satisfied. (implicitly differentiate, and show that this differentiation is the same as the original DE)

Question 15

What is a solution curve for a given DE?

Answer 15

The graph of a solution ϕ of an ODE over the domain of the solution I is called a solution curve.

Question 16

What is a family of solutions?

Answer 16

An n-parameter family of (explicit) solutions to an ODE is a real valued function of the form y = …, such that any values of any of the constants results in a solution to the given DE on I.

Question 17

What is a particular solution to a DE?

Answer 17

A solution of an ODE that is free of arbitrary parameters (no constants)

Question 18

What is a singular solution to a DE?

Answer 18

A solution of an ODE that cannot be obtained by specializing any of the parameters in a family of solutions is called a singular solution.

Question 19

What is a General Solution to a DE?

Answer 19

An n-parameter family of solutions such that every solution of the given DE can be obtained by varying the values of the constants on the interval I.

Question 20

What is an Initial-value problem?

Answer 20

On some interval I containing x0 the problem of solving an nth order ODE subject to n side conditions specifying the values of y and its first n1 derivatives at x0 is called an nth order initial-value problem (IVP). The specified values are called initial conditions

Question 21

What is a Boundary-value Problem?

Answer 21

A DE that is subject to conditions specified on y(x) and its derivatives at two (or more) different points (called boundary conditions) These boundary conditions along with the DE make a boundary value problem. (an IVP with multiple conditions that can occur anywhere on y(x))

Question 22

What is yc?

Answer 22

Yc is the complimentary solution to the given DE. It is the solution to the associated homogenous DE on the interval I.

Question 23

What is yp?

Answer 23

Yp is a particular solution to the given DE. It is a solution without constants to the given DE on the interval I.

Question 24

What is a Taylor Series?

Answer 24

Look up