Midterm Concepts

Question 1

A derivative may be involved implicitly through the presence of:

Answer 1

differentials

Question 2

When an equation involves one or more derivatives with respect to a
particular variable, that variable is called an:

Answer 2

independent variable

Question 3

A variable
is called _______ if a derivative of that variable occurs.

Answer 3

dependent

Question 4

The _____ of a differential equation is the order of the highest-ordered
derivative appearing in the equation.

Answer 4

order

Question 5

F(x, y, y’, .. . , y<nl) = 0 is called an ______ ordinary differential equation.

Answer 5

“nth-order”

Question 6

Applications Leading to Differential Equations

Answer 6

Exponential Growth – Population and Exponential Reduction or Decay

Question 7

Is an equation that contains one or more functions with its
derivatives

Answer 7

differential equation

Question 8

The ______ of the function define the rate of change of a function at a point.

Answer 8

derivatives

Question 9

An equation involving the independent variable x, dependent variable y and the differential coefficients of
dependent variable with respect to independent variable.

Answer 9

differential equation

Question 10

An equation which contains one or more terms and the derivatives of one variable (i.e., dependent variable)
with respect to the other variable (i.e., independent variable)

Answer 10

differential equation

Question 11

Is one which contain at least one derivative

Answer 11

differential equation

Question 12

Is the same as the order of the highest-ordered derivative in the equation.

Answer 12

order of a D.E

Question 13

Is the same as the power (or exponent) to which the derivative of the highest order in
the equation is raised.

Answer 13

degree of a D.E

Question 14

The most general solutions of a D.E. of the nth order contains an _______

Answer 14

arbitrary constants.

Question 15

If particular values
are assigned to these arbitrary constants, ;the solution is called

Answer 15

particular solution

Question 16

Involves function and its derivatives. It contains only one
independent variable and one or more of its derivatives with respect to the variable

Answer 16

Ordinary differential equation

Question 17

Is a differential equation containing partial derivatives of the
dependent variable (one or more) with more than one independent variable.

Answer 17

Partial differential equation

Question 18

The degree of any differential
equation can be found when it is in the form a ______; otherwise, the degree cannot be defined.

Answer 18

polynomial

Question 19

𝒂𝒏(x)𝒚
(𝒏) + 𝒂𝒏−𝟏(x)𝒚
(𝒏−𝟏) + … + 𝒂𝟏(x)𝒚
′ + 𝒂𝟎(x)y = g(x)

Here, ______ represent functions of x, possibly constants, that are attached to y and its derivatives by
multiplication.

Answer 19

𝒂𝒏 𝐱

Question 20

The term g(x) is not attached to y or its derivatives by multiplication and may be a function of
x, or just a constant, possibly 0. The term g(x) is called a ______

Answer 20

forcing function

Question 21

A _______ does not combine y or any of its derivatives through multiplication.

Answer 21

linear differential equation

Question 22

Non-linear ODE contains functions of one of the dependent variable on its derivatives such as:

Answer 22

Trigonometric functions, Exponential functions, Logarithmic functions

Question 23

A linear differential equation is ______ if the forcing function g(x) is 0.

Answer 23

homogenous

Question 24

Homogeneity only applies to _____ differential equations

Answer 24

LINEAR

Question 25

Any ________ between the variables of a differential equation which satisfies the equation is a solution of it

Answer 25

non derivative relation

Question 26

The graph of the solutions of a differential equation are called its:

Answer 26

integral curves

Question 27

A solution of an ordinary differential equation of order n that involves exactly n essential arbitrary constants

Answer 27

General Solution

Question 28

General Solution is also called:

Answer 28

complete solution, general integral.

Question 29

A solution of a partial differential equation that involves arbitrary functions.

Answer 29

General Solution

Question 30

Is a solution obtained from the General Solution by assigning specific values to the arbitrary constants.

Answer 30

Particular solution

Question 31

A solution ______ of a differential equation that contains no arbitrary constants is called a particular solution to the equation.

Answer 31

yp(x)

Question 32

Sets the initial condition of the problem, normally at zero value of the independent variable, to obtain the value of the arbitrary constant.

Answer 32

Initial Value Ploblem

Question 33

Originated in problems of motion where the independent variable is t (representing elapsed time), and the initial conditions are the position and velocity of an object at the initial (starting) time of an experiment.

Answer 33

initial value problem

Question 34

The term initial condition is used when the independent variable is the time and the condition given are at:

Answer 34

zero -time

Question 35

Is used if the independent variable is anyone of the spatial coordinates x, y or z and the conditions given are for known values of anyone of these coordinates

Answer 35

term boundary

Question 36

The value of the _______ can only be determined with the help of the IVP present with the differential equation

Answer 36

arbitrary constant(s)

Question 37

The _______ varies according to the directional field of the problem.

Answer 37

arbitrary constant

Question 38

Is a result of the magnitude (dependent) of the function observed at any given time (independent).

Answer 38

arbitrary constant

Question 39

The graph of a solution of a differential equation is a:

Answer 39

solution curve

Question 40

The order of a differential equation to be solved _____ to the number of arbitrary constants added to the solution

Answer 40

equates

Question 41

The number of derivatives of the produced differential equation should be ______ to the number of distinct arbitrary constants to be eliminated

Answer 41

equal

Question 42

Is defined as those polynomials all of those terms are of the same degree.

Answer 42

Homogenous polynomials

Question 43

The _______ that appear the same in every term of the variables x and y is said to be the degree of homogenous equation.

Answer 43

constant exponent

Question 44

it is possible to transform the non-exact differential equation by a judicious multiplication which we call as:

Answer 44

integrating factor

Question 45

An ______ is a solution to a certain partial differential equation.

Answer 45

integrating factor

Question 46

______ is a generalized case of the linear form

Answer 46

Bernoulli Equation