Midterm Concepts Flashcards
A derivative may be involved implicitly through the presence of:
differentials
When an equation involves one or more derivatives with respect to a
particular variable, that variable is called an:
independent variable
A variable
is called _______ if a derivative of that variable occurs.
dependent
The _____ of a differential equation is the order of the highest-ordered
derivative appearing in the equation.
order
F(x, y, y’, .. . , y<nl) = 0 is called an ______ ordinary differential equation.
“nth-order”
Applications Leading to Differential Equations
Exponential Growth – Population and Exponential Reduction or Decay
Is an equation that contains one or more functions with its
derivatives
differential equation
The ______ of the function define the rate of change of a function at a point.
derivatives
An equation involving the independent variable x, dependent variable y and the differential coefficients of
dependent variable with respect to independent variable.
differential equation
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)
differential equation
Is one which contain at least one derivative
differential equation
Is the same as the order of the highest-ordered derivative in the equation.
order of a D.E
Is the same as the power (or exponent) to which the derivative of the highest order in
the equation is raised.
degree of a D.E
The most general solutions of a D.E. of the nth order contains an _______
arbitrary constants.
If particular values
are assigned to these arbitrary constants, ;the solution is called
particular solution
Involves function and its derivatives. It contains only one
independent variable and one or more of its derivatives with respect to the variable
Ordinary differential equation
Is a differential equation containing partial derivatives of the
dependent variable (one or more) with more than one independent variable.
Partial differential equation
The degree of any differential
equation can be found when it is in the form a ______; otherwise, the degree cannot be defined.
polynomial
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.
𝒂𝒏 𝐱
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 ______
forcing function
A _______ does not combine y or any of its derivatives through multiplication.
linear differential equation
Non-linear ODE contains functions of one of the dependent variable on its derivatives such as:
Trigonometric functions, Exponential functions, Logarithmic functions
A linear differential equation is ______ if the forcing function g(x) is 0.
homogenous
Homogeneity only applies to _____ differential equations
LINEAR
Any ________ between the variables of a differential equation which satisfies the equation
is a solution of it
non derivative relation
The graph of the solutions of a differential equation are called its:
integral curves
A solution of an ordinary differential equation of order n that involves exactly n essential arbitrary constants
General Solution
General Solution is also called:
complete solution, general integral.
A solution of a partial differential equation that involves arbitrary functions.
General Solution
Is a solution obtained from the General Solution by assigning
specific values to the arbitrary constants.
Particular solution
A solution ______ of a differential equation that contains no arbitrary constants is called a particular solution to
the equation.
yp(x)
Sets the initial condition of the problem, normally at zero value of the independent variable, to obtain the value
of the arbitrary constant.
Initial Value Ploblem
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.
initial value problem
The term initial condition is used when the independent variable is the time and the condition given are at:
zero -time
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
term boundary
The value of the _______ can only be determined with the help of the IVP present with the
differential equation
arbitrary constant(s)
The _______ varies according to the directional field of the problem.
arbitrary constant
Is a result of the magnitude (dependent) of the function observed at any
given time (independent).
arbitrary constant
The graph of a solution of a differential equation is a:
solution curve
The order of a differential equation to be solved _____ to the number of arbitrary constants added to
the solution
equates
The number of derivatives of the produced differential equation should be ______ to the number of distinct
arbitrary constants to be eliminated
equal
Is defined as those polynomials all of those terms are of the same degree.
Homogenous polynomials
The
_______ that appear the same in every term of the variables x and y is said to be the degree of
homogenous equation.
constant exponent
it is possible to
transform the non-exact differential equation by a judicious multiplication which we call as:
integrating factor
An ______ is a solution to a certain partial differential equation.
integrating factor
______ is a generalized case of the linear form
Bernoulli Equation
