1. The Structure Theorem: General Solution = Homogeneous Family + One Particular Solution
We study the second-order linear non-homogeneous ODE with constant coefficients:
$$a\,y'' + b\,y' + c\,y = f(x) \qquad (a \neq 0)$$where $a, b, c$ are real constants and $f(x) \neq 0$ is the forcing function (also called the right-hand side or the non-homogeneous term).
Structure Theorem (General Solution)
The general solution of $\;a\,y'' + b\,y' + c\,y = f(x)\;$ is:
$$\boxed{y = y_h + y_p}$$where
- $y_h = c_1 y_1 + c_2 y_2$ is the general solution of the corresponding homogeneous equation $a\,y'' + b\,y' + c\,y = 0$, and
- $y_p$ is any one particular solution of the non-homogeneous equation.
Why does this work? — Linearity's Gift
Suppose $y_A$ and $y_B$ are both solutions of the non-homogeneous equation. Define $w = y_A - y_B$. Then:
$$a\,w'' + b\,w' + c\,w = a\,y_A'' + b\,y_A' + c\,y_A - \bigl(a\,y_B'' + b\,y_B' + c\,y_B\bigr) = f(x) - f(x) = 0$$So $w = y_A - y_B$ satisfies the homogeneous equation. In other words, the difference of any two solutions of the non-homogeneous equation is a solution of the homogeneous equation. This means every solution of the non-homogeneous equation has the form $y_h + y_p$ for some choice of the constants $c_1, c_2$.
Recall from first-order linear equations $y' + p(x)y = q(x)$: the general solution was $y = y_h + y_p$, where $y_h = Ce^{-\int p\,dx}$ and $y_p$ was found by an integrating factor. The same structural pattern now carries over to second order — it is the arc's load-bearing principle.
The Task Split
Find $y_h$: solve $a\,y'' + b\,y' + c\,y = 0$ via the characteristic equation $ar^2 + br + c = 0$. This was the previous station's business (distinct real roots, repeated roots, complex roots).
Find one particular solution $y_p$ of $a\,y'' + b\,y' + c\,y = f(x)$. Two methods: Undetermined Coefficients and Variation of Parameters.