A linear ODE has the form $y' + P(x)y = Q(x)$ where $P$ and $Q$ depend only on $x$, not $y$. Option (C): $y' + xy = e^x y \Rightarrow y' + (x-e^x)y = 0$ — this IS linear (coefficient of $y$ is $x-e^x$, a function of $x$ only). Option (B): $y' = (\sin y)x + e^x$ has $\sin y$ — nonlinear in $y$. Option (A): $y' - (\sin x)y = 0$ — linear. Option (D): $y' = 5$ — linear.