[Math] a particular and a homogenous solution of a differential equation

Question

When solving linear nonhomogeneous equations, we deal with two types of solutions:

• particular
• homogeneous

Why do we have these two types of solutions for differential equations? What does each of them represent?

Answer 1

A linear differential equation can be expressed as $D f = g$, where $D$ is some linear operator on functions built from differentiation, and $g$ is an arbitrary function. A particular solution is a function $f$ that satisfies that equation. But note that if $f_1$ and $f_2$ are two particular solutions, then $D(f_1 - f_2) = Df_1 - Df_2 = g - g = 0$. That is, the difference between any two particular solutions is a solution of the homogenous equation $Df = 0$.

It is usually much easier to solve the homogenous equation than the original equation. So if you want to find all particular solutions to the original equation, it suffices to find one solution to it, and all solutions to the homogenous equation. Then sums of the single particular solution and each of the homogenous solutions gives all the particular solutions.

A very simple example: Consider $f''(x) = x$. One particular solution is $f(x) = {x^3\over 6}$. The homogenous equation is $f''(x) = 0$, whose general solution is $f(x) = Ax + B$, for various values of $A, B$. Thus the general solution for the equation $f''(x) = x$ is $$f(x) = {x^3\over 6} + Ax + B$$