Chapter 4 - Vector Spaces - 4.3 Subspaces - Problems - Page 273: 30

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Let $F(x)$ be a non-zero function on an interval $I$ We can write set $S$ as $S=\{S \in C^2(I): y''+a_1y'+a_2y=F(x)\}$. Assume $y_0 \in C^2(I)$ such as $y_0(x)=0 \forall x \in I$ Then $y_0''(x)=0,y'_0(x)=0 \forall x \in I$ Obtain $y''_0(x)+a_1y'_0(x)+a_2y_0(x)=0 \forall x \in I$ Since $F(x_0) \ne 0 \rightarrow y''_0(x_0)+a_1y'_0(x)+a_2y_0(x)=0 \ne F(x_0)$ $y_0 \notin S$, hence $S$ is not a subspace of $C^2(I)$