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In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that ...
In solving mathematical equations, particularly linear simultaneous equations, differential equations and integral equations, the terminology homogeneous is often used for equations with some linear operator L on the LHS and 0 on the RHS. In contrast, an equation with a non-zero RHS is called inhomogeneous or non-homogeneous, as exemplified by ...
In the case of a first order ODE that is non-homogeneous we need to first find a solution to the homogeneous portion of the DE, otherwise known as the associated homogeneous equation, and then find a solution to the entire non-homogeneous equation by guessing.
Consider a linear non-homogeneous ordinary differential equation of the form = + (+) = where () denotes the i-th derivative of , and denotes a function of .. The method of undetermined coefficients provides a straightforward method of obtaining the solution to this ODE when two criteria are met: [2]
A linear differential equation that fails this condition is called inhomogeneous. A linear differential equation can be represented as a linear operator acting on y(x) where x is usually the independent variable and y is the dependent variable. Therefore, the general form of a linear homogeneous differential equation is =
Intuitively, one can think of the inhomogeneous problem as a set of homogeneous problems each starting afresh at a different time slice t = t 0. By linearity, one can add up (integrate) the resulting solutions through time t 0 and obtain the solution for the inhomogeneous problem. This is the essence of Duhamel's principle.
In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; the power is called the degree of homogeneity, or simply the degree.
The only difference between a homogeneous and an inhomogeneous (partial) differential equation is that in the homogeneous form we only allow 0 to stand on the right side ((,) =), while the inhomogeneous one is much more general, as in (,) could be any function as long as it's continuous and can be continuously differentiated twice.