Differential Equation of Family of Linear Combination of Functions is Linear

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

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

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

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

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

which leaves:
 * $\dfrac {\mathrm d y} {\mathrm d x}- \dfrac {f' \left({x}\right)} {f \left({x}\right)} y = g \left({x}\right) \left({1 - \dfrac {f' \left({x}\right)} {f \left({x}\right)} }\right)$

which is linear and of first order.