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 $(2)$:
\(\ds \dfrac {\d y} {\d x}\) | \(=\) | \(\ds \dfrac {y - \map g x} {\map f x} \map {f'} x + \map {g'} x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\map {f'} x} {\map f x} y - \dfrac {\map g x \map {f'} x} {\map f x} + \map g x\) |
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 $6$