Differential Equation of Family of Linear Combination of Functions is Linear

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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 $(1)$ with respect to $x$ 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 $(2)$:

\(\displaystyle \dfrac {\d y} {\d x}\) \(=\) \(\displaystyle \dfrac {y - \map g x} {\map f x} \map {f'} x + \map {g'} x\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\map {f'} x} {\map f x} y - \dfrac {\map g x \map {f'} x} {\map f x} + \map g x\)


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.

$\blacksquare$


Sources