# 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

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 2.10$: Problem $6$