Question

Let \(C(\mathbb{R})\) be the linear space of all continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\).

a) Let \(S_{c}\) be the set of differentiable functions \(u(x)\) that satisfy the differential equation
\[
u^{\prime}=2 x u+c
\]
for all real \(x\). For which value(s) of the real constant \(c\) is this set a linear subspace of \(C(\mathbb{R})\) ?
b) Let \(C^{2}(\mathbb{R})\) be the linear space of all functions from \(\mathbb{R}\) to \(\mathbb{R}\) that have two continuous derivatives and let \(S_{f}\) be the set of solutions \(u(x) \in C^{2}(\mathbb{R})\) of the differential equation
\[
u^{\prime \prime}+u=f(x)
\]
for all real \(x\). For which polynomials \(f(x)\) is the set \(S_{f}\) a linear subspace of \(C(\mathbb{R})\) ?
c) Let \(\mathcal{A}\) and \(\mathcal{B}\) be linear spaces and \(L: \mathcal{A} \rightarrow \mathcal{B}\) be a linear map. For which vectors \(y \in \mathcal{B}\) is the set
\[
\mathcal{S}_{y}:=\{x \in \mathcal{A} \mid L x=y\}
\]
a linear space?