Superposition principle

Statement

Let $I$ be an open interval, and let $L y = y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y$ with continuous functions $a_0, \dots, a_{n-1}$ on $I$. Let $y_1$ and $y_2$ be solutions of the homogeneous linear equation $L y = 0$ on $I$, and let $c_1, c_2$ be real numbers. Then $c_1 y_1 + c_2 y_2$ is a solution of $L y = 0$ on $I$.

Remarks

The proof is the linearity of $L$: $L(c_1 y_1 + c_2 y_2) = c_1 L y_1 + c_2 L y_2 = 0$. The same argument gives a variant for right sides: if $L y_1 = b_1$ and $L y_2 = b_2$, then $L(y_1 + y_2) = b_1 + b_2$. The principle fails for nonlinear equations. For $y' = y^2$, the function $y = -1/x$ is a solution on $x > 0$, but $2y$ is not.

Explanations

L1 human

For a linear equation, you can add solutions and scale them, and you get solutions again. In physics this is superposition: two waves that solve a linear wave equation add up to a third solution.