Notes of ODE

8/3/2019 Notes of ODE

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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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Notes of ODE