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General Terms of ODE
Ordinary Differential Equations
Equation: Equations describe the relations between the dependent and independent
variables. An equal sign "=" is required in every equation.
Differential Equation: Equations that involve dependent variables and their derivativeswith respect to the independent variables are called differential equations.
Ordinary Differential Equation: Differential equations that involve only ONEindependentvariable are called ordinarydifferential equations.
Partial Differential Equation: Differential equations that involve twoormoreindependentvariables are called partialdifferential equations.
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Order and Degree
Order: The order of a differential equation is the highestderivative that appears in thedifferential equation.
Degree: The degree of a differential equation is the power of the highestderivative term.
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Linear, Non-linear, and Quasi-linear
Linear: A differential equation is called linear if there are no multiplications amongdependent variables and their derivatives. In other words, all coefficients are functions ofindependent variables.
Non-linear: Differential equations that do not satisfy the definition of linear are non-linear.
Quasi-linear: For a non-linear differential equation, if there are no multiplications amongall dependent variables and their derivatives in the highest derivative term, the differentialequation is considered to be quasi-linear.
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Homogeneous
Homogeneous: A differential equation is homogeneous if every single term contains thedependent variables or their derivatives.
Non-homogeneous: Differential equations which do not satisfy the definition ofhomogeneous are considered to be non-homogeneous.
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Solutions
General Solution: Solutions obtained from integrating the differential equations are called
general solutions. The general solution of a order ordinary differential equation contains
arbitrary constants resulting from integrating times.
Particular Solution: Particular solutions are the solutions obtained by assigning specificvalues to the arbitrary constants in the general solutions.
Singular Solutions: Solutions that can not be expressed by the general solutions arecalled singular solutions.
Conditions
Initial Condition: Constrains that are specified at the initial point, generally time point, arecalled initial conditions. Problems with specified initial conditions are called initial valueproblems.
Boundary Condition: Constrains that are specified at the boundary points, generally spacepoints, are called boundary conditions. Problems with specified boundary conditions arecalled boundary value problems.
First Order ODE
Definition
First orderordinary differential equations have the general form of
or
Special First Order Ordinary Differential Equations
Although the above general forms look simple, there is no single rule to solve them. Somespecial cases are categorized as follows and their solutions or solving methods can be foundby clicking the category names:
Differential Equation Format
Separable differential equation
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;
Exact differential equation
Linear differential equation
Implicit differential equation
Separable Differential Equations
Differential Equation General Solution/Simplifying Method
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Exact Differential Equations
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Differential Equation General Solution
Same as the above multiplied by the integrating
factor .
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Integrating Factors
In solving "exactable" ordinary differential equations, the following table of common exactdifferential forms may help.
Exact Differential Form Integrating Factor
-
-
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The above table also shows that the integrating factors for a given exact differential formare notunique.
Linear Differential Equations
Linear differential equations and some special quasi-linear differential equations which can
be linearized are listed in the following table. Their solutions or simplifying methods are alsopresented.
Differential Equation General Solution/Simplifying Method
Linear differential equation:
where is the integrating factor.
Bernoulli's differential equation:
which is a linear differential equation.
Ricatti's differential equation:
which is the Bernoulli's differential equation
with n=2 and a non-homogeneous term.
where is the particular solution
Existence and Uniqueness of Solutions
Consider a first order differential equation
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with the initial condition
where is bounded in the neighborhood of the initial point, i.e.,
in
Sufficient Condition of Existence: If is continuous in the neighborhood region
, the solution of this initial value problem in the region exists.
Sufficient Condition of Existence and Uniqueness: If and its partial derivative with
respect to are continuous in the neighborhood region , the solution ofthis initial value problem in the region exists and is unique.
Picard Iteration Method: The unique solution of the above initial value problem is
where
Higher Order Linear ODE
Linear Ordinary Differential Equations
Linear Ordinary Differential Equations: A orderlinearordinary differential equationshave the general form of
where are all functions of .
This differential equation is homogeneous if . Otherwise, it is a non-homogeneousdifferential equation.
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Linear Dependence and Independence
Linear Dependence: Consider a set of functions defined on
. If there exist constants which satisfy the following two conditions,
then these functions are called linearly dependant on .
In other words, if one of these functions can be expressed in terms (by linear combination)
of others, these functions are linearly dependent on the interval .
Linear Independence: Consider a set of functions defined on
. If the only way to make the linear combination of these functions be zero is that
all constants are zero , this set of functions is called linearly
independent on .
In other words, if none of these functions can be expressed in terms (by linear combination)
of others, these functions are linearly independent on the interval .
The Wronskian: Consider a set of functions differentiable to
the order on . The Wronskian of this set of function is
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where is the determinant.
If the Wronskian is zero, this set of functions is linearly dependent. Ifnotzero, this set is
linearly independenton .
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Solutions and Superposition
Linear Combination ofSolutions: Consider a orderlinearhomogeneousordinarydifferential equations
If are solutions of this linear homogeneous differential equation,their linear combinations are also solutions of this equation, i.e.,
where .
General Solutions ofLinear Homogeneous Differential Equations: Consider aorderlinearhomogeneousordinary differential equations
If are n independentsolutions of this differential equation, their linearcombinations form the general solution of this equation, i.e.,
where are arbitrary constants.
Particular Solutions: Consider a orderlinearnon-homogeneousordinary differentialequations
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where .
If contains no arbitrary constants and satisfies this differential equation, i.e.,
is called the particular solution of this equation.
General Solutions ofLinear Non-homogeneous Differential Equations: Consider a
orderlinearnon-homogeneousordinary differential equations
where .
If is the particular solution
and , the complementary solution, is the general solution of the associatedhomogeneous differential equation
then the general solution of the linear non-homogeneous equation is the superposition ofboth particular and complementary solutions
where are arbitrary constants, are n independent
solutions of the associated homogeneous equation.Form
A linearhomogeneousordinary differential equation with constant coefficients has thegeneral form of
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where are all constants.
2nd Order Linear Homogeneous ODE with Constant Coefficients
A second order linear homogeneous ordinary differential equation with constantcoefficientscan be expressed as
This equation implies that the solution is a function whose derivatives keep the same formas the function itself and do not explicitly contain the independent variable , sinceconstant coefficients are not capable of correcting any irregular formats or extra variables.
An elementary function which satisfies this restriction is the exponential function .
Substitute the exponential function into the above differential equation, thecharacteristic equation of this differential equation is obtained
This characteristic equation has two roots and .
2nd Order Linear Homogeneous ODE with Constant Coefficients:
Characteristic Equation:
Solutions of Characteristic Equation , General Solution
1
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2
3
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nth Order Linear Homogeneous ODE with Constant Coefficients
Similar to the second order equations, the form, characteristic equation, and general
solution of order linear homogeneous ordinary differential equations are summarized asfollows:
nth Order Linear Homogeneous ODE with Constant Coefficients:
Characteristic Equation:
Solutions of Characteristic
Equation
General Solution
1 are all different
real numbers.
2are k repeated
real roots; others are different
real numbers.
3
are k/2 pairs of
complex conjugate roots
; others are different
real numbers.
Standard Form
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A linearnon-homogeneousordinary differential equation with constant coefficients has thegeneral form of
where are all constants and .
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Particular Solutions
For a linear non-homogeneous differential equation, the general solution is the
superposition of the particular solution and the complementary solution .
See further discussion
The complementary solution which is the general solution of the associated
homogeneous equation ( ) is discussed in the section ofLinear Homogeneous ODE
with Constant Coefficients. This section summarizes common methodologies on solving the
particular solution .
Method of Undetermined Coefficients: The non-homogeneous term in a linear non-homogeneous ODE sometimes contains only linear combinations or products of some simplefunctions whose derivatives are more predictable or well known. By understanding thesesimple functions and their derivatives, we can guessthetrialsolutionwith undeterminedcoefficients, plug into the equation, and then solve for the unknown coefficients to obtainthe particular solution. This method is called the method of undetermined coefficients.(See further detail.)
Method of Variation ofParameters: If the complementary solution has been found in alinear non-homogeneous ODE, one can use this complementary solution and vary thecoefficients to unknown parameters to obtained the particular solutions. This methods iscalled the method of variation ofparameters. (See further detail.)
Method of Reduction of Order: When solving a linear homogeneous ODE with constantcoefficients, we factor the characteristic equation to obtained the homogeneous solution.
Similarly, the method of reduction of order factors the differential operatorsand inverses (integrates) them one by one to reduce the order and eventually obtain the
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particular solution. (See further detail.)
Method of Inverse Operators: The method of inverse operators takes a step furtherthan the method of reduction of order by categorizing how the inverse differential operator
and its higher order operators affect common functions to achieve a more systematic
way to obtain the particular solution. (See further detail.)
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