|Linear Ordinary Differential Equations|
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|
If are solutions of this linear homogeneous differential equation, their linear combinations are also solutions of this equation, i.e.,
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.
If contains no arbitrary constants and satisfies this differential equation, i.e.,
is called the particular solution of this equation.
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.