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 nonhomogeneous 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 Nonhomogeneous Differential Equations: Consider a order linear nonhomogeneous 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 nonhomogeneous equation is the superposition of both particular and complementary solutions
where are arbitrary constants, are n independent solutions of the associated homogeneous equation.
