Differential Equation of Family of Linear Combination of Functions is Linear

Theorem
Consider the one-parameter family of curves:
 * $(1): \quad y = C \map f x + \map g x$

The differential equation that describes $(1)$ is linear and of first order.

Proof
Differentiating $(1)$ $x$ gives:
 * $(2): \quad \dfrac {\d y} {\d x} = C \map {f'} x + \map {g'} x$

Rearranging $(1)$, we have:
 * $C = \dfrac {y - \map g x} {\map f x}$

Substituting for $C$ in $(2)$:

which leaves:
 * $\dfrac {\d y} {\d x} - \dfrac {\map {f'} x} {\map f x} y = \map g x \paren {1 - \dfrac {\map {f'} x} {\map f x} }$

which is linear and of first order.