A differential equation is a series of (0 .... n)th-order derivatives (or partial derivatives), each multiplied by a function of the independent variable(s), added together and set equal to another function of the independent variables. For example,
p(t)*y" + q(t)*y' + r(t)*y = g(t)
where y = f(t). This is equally the case for partial differential equations, where p, q, r, g, and f are functions of more than one variable, and all the derivatives are partials.
A differential equation is said to be homogeneous if g(t) (or g(x, y, z, t, ...... )) is 0, e.g., p(t)*y" + q(t)*y' + r(t)*y = 0. When this is not the case, the equation is called inhomogeneous (sometimes nonhomogeneous).
A system of differential equations is said to be homogeneous if all gn(x, y, z, t, ... ) = 0.
For every homogeneous equation, there is an infinite number of inhomogeneous equations. This makes the solution of differential equations somewhat less painful, because once the solution to the homogeneous equation has been found, it is always possible to find the solution to a corresponding inhomogeneous equation. Thus, the solution of homogeneous equations is fundamental to differential mathematics.
Homogeneous differential equations are a subset of linear differential equations, i.e., all homogeneous equations are linear.
The general algorithm for solving an ordinary linear homogeneous differential equation is:
- Find the characteristic equation.
- Use given initial conditions to solve for the constants.
- Replace the constants with these values in the characteristic equation.
Partial linear homogeneous differential equations are a little harder.