General solution of an inhomogeneous linear equation

Statement

Let $I$ be an open interval, let $a_0, \dots, a_{n-1}, b : I \to \mathbb{R}$ be continuous, and let $L y = y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y$. Let $y_p$ be a solution of $L y = b$ on $I$, and let $(y_1, \dots, y_n)$ be a fundamental system of $L y = 0$ on $I$. Then a function $y : I \to \mathbb{R}$ is a solution of $L y = b$ on $I$ if and only if $$y = y_p + c_1 y_1 + \dots + c_n y_n$$ for some real numbers $c_1, \dots, c_n$.

Remarks

Proof: $y$ solves $L y = b$ if and only if $L(y - y_p) = 0$, by linearity. Then $y - y_p$ is a combination of the fundamental system. The method of undetermined coefficients and variation of parameters find a particular solution $y_p$. Example: $y'' + y = 1$ has $y_p = 1$, so its solutions are $1 + c_1 \cos x + c_2 \sin x$.

Explanations

L1 human

To solve a linear equation with a right side, find one solution by any method. Then add all the solutions of the equation without right side. This gives every solution, and each only one time.