Proof of the superposition principle

Proof

1. Let $u = c_1 y_1 + c_2 y_2$. As solutions of the homogeneous linear equation, $y_1$ and $y_2$ are $n$ times differentiable on $I$, with $L y_1 = 0$ and $L y_2 = 0$.
2. We show by induction on $k$ that, for $0 \le k \le n$, $u$ is $k$ times differentiable on $I$ and $u^{(k)} = c_1 y_1^{(k)} + c_2 y_2^{(k)}$.
3. For $k = 0$ this is the definition of $u$.
4. Let $k < n$ and assume the claim for $k$. The functions $y_1^{(k)}$ and $y_2^{(k)}$ are differentiable, since $k < n$.
5. By linearity of differentiation, $u^{(k)} = c_1 y_1^{(k)} + c_2 y_2^{(k)}$ is differentiable with derivative $c_1 y_1^{(k+1)} + c_2 y_2^{(k+1)}$. This proves the claim for $k + 1$.
6. Write $a_n = 1$. By step 2, at each $x \in I$, $L u = \sum_{k=0}^{n} a_k \left(c_1 y_1^{(k)} + c_2 y_2^{(k)}\right) = c_1 L y_1 + c_2 L y_2$.
7. By step 1, this is $c_1 \cdot 0 + c_2 \cdot 0 = 0$. So $u$ is $n$ times differentiable and solves the linear equation $L y = 0$ on $I$.