Higher Order Linear Ordinary Differential Equations and Solutions
engineering fundamentals Higher Order
Linear Ordinary Differential Equations

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Linear Ordinary Differential Equations

Linear Ordinary Differential Equations: A order linear ordinary differential equations have the general form of

where are all functions of .

This differential equation is homogeneous if . Otherwise, it is a non-homogeneous differential equation.

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

where is the determinant.

If the Wronskian is zero, this set of functions is linearly dependent. If not zero, this set is linearly independent on .

Solutions and Superposition

Linear Combination of Solutions: Consider a order linear homogeneous ordinary differential equations

If are solutions of this linear homogeneous differential equation, their linear combinations are also solutions of this equation, i.e.,

where .

General Solutions of Linear Homogeneous Differential Equations: Consider a order linear homogeneous ordinary differential equations

If are n independent solutions of this differential equation, their linear combinations form the general solution of this equation, i.e.,

where are arbitrary constants.

Particular Solutions: Consider a order linear non-homogeneous ordinary differential equations

where .

If contains no arbitrary constants and satisfies this differential equation, i.e.,

is called the particular solution of this equation.

General Solutions of Linear Non-homogeneous Differential Equations: Consider a order linear non-homogeneous ordinary differential equations

where .

If is the particular solution

and , the complementary solution, is the general solution of the associated homogeneous differential equation

then the general solution of the linear non-homogeneous equation is the superposition of both particular and complementary solutions

where are arbitrary constants, are n independent solutions of the associated homogeneous equation.

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