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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?

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1 Answer 1

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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 $$f(x) = {x^3\over 6} + Ax + B$$

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  • $\begingroup$ Can you expand on the first paragraph why we are done after showing that the two particular solutions are a solution to the homogenous equation? $\endgroup$
    – mdcq
    Commented Jan 18, 2018 at 15:20
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    $\begingroup$ @philmcole - (1) I showed that the difference of two particular solutions is a homogenous solution. The particular solutions themselves are not solutions to the homogenous equation. (2) if $f_1, f_2$ are particular solutions, then $f_2 = f_1 + (f_2 - f_1)$, and $(f_2 - f_1)$ is a homogenous solution. So if I know $f_1$, then $f_2$ must be among all the functions of form $f_1 + h$ for some homogenous solution $h$. $\endgroup$ Commented Jan 18, 2018 at 16:06
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    $\begingroup$ Thanks! Does this show also that $f_1 + f_h$, where $f_h$ is the general homogeneous solution, is indeed the general solution and there are no other ones? $\endgroup$
    – mdcq
    Commented Jan 18, 2018 at 17:02

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