Let S be the set of all infinitely differentiable functions whose domain and range are the real number. Let $S_1, S_2, S_3, $ and $S_4$ be the fours subsets of the S defined below:
$S_1$ = The set of infinity differentiable functions $f(x)$ such that $ \int_{0}^{3} f(x) dx = 0 $
$S_2$ = The set of infinity differentiable functions $f(x)$ such that $ f''(x) + 2\ln(x)f'(x) = \sin(x)f(x) $
$S_3$ = The set of infinity differentiable functions $f(x)$ such that $f(x)$ is either always increasing, always decreasing, or $f(x) = 0 $
$S_4$ = The set of infinity differentiable functions $f(x)$ such that $ (f''(x))^2 = 2f(x)$
How many of the above subsets satisfy both of the following conditions?
- Given any two functions $f_1(x) $ and $f_2(x)$ in the subset, the function $f_1(x)+ f_2(x)$ is also in the subset.
- Given any function $f(x)$ in the subset and any real number $c$, the function $cf(x)$ is also in the subset.
You can either guide me to the answer or provide the answer. Whichever would provide the most intuitive solution.