I am noting two important theorems which describe the formal form of a general solution of a homogenous linear nth-order differential equation. Let your equation is as: $a_n(x)y^{(n)}+a_{n-1}(x)y^{(n-1)}+...+a_1(x)y'+a_0(x)y=0$ where the functions $a_i(x), 1\leq i \leq n$ and $g(x)$ be continuous on an interval $I$ and $a_n(x)\neq0 $ over the interval. Then:
Theorem 1: There exists a linearly independent solutions, called fundamental set of solutions, for above equation on interval $I$ as: $y_1,y_2,...y_n$
Theorem 2. if $y_1,y_2,...y_n$ be a fundamental set of solutions of above linear and homogenous nth-order equation on interval $I$. Then the general solution of the equation on $I$ is defined to be: $y=c_1y_1(x)+c_2y_2(x)+...+c_ny_n(x)$ wherein $c_i, 1\leq i\leq n$ are arbitrary constants.
Now, I think you can conclude why the general solution of your first order linear and homogenous equation is as $y=Cf(x)$