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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?
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