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The armour formula is a mathematical formula used to solve first-order inhomogeneous differential equations. The formula is named the armour formula as it is able to solve all first-order differential equations.

The formula $$y=e^{-F(x)} \int e^{F(x)} g(x) \,\mathrm{d}x$$ is the general solution to all first-order differential equations on the form $$\frac{\mathrm{d}y}{\mathrm{d}x}+f(x)y=g(x)$$.

Proof
$$\begin{align} \frac{\mathrm{d}y}{\mathrm{d}x}+f(x)\cdot y&=g(x) & &\textrm{(1)} \\ e^{F(x)}\cdot \frac{\mathrm{d}y}{\mathrm{d}x}+f(x)\cdot e^{F(x)}\cdot y&=e^{F(x)}\cdot g(x) & &\textrm{(2)} \\ \frac{\mathrm{d}}{\mathrm{d}x}(e^{F(x)}\cdot y)&=e^{F(x)}\cdot g(x) & &\textrm{(3)} \\ e^{F(x)}\cdot y&=\int e^{F(x)}\cdot g(x)\,\mathrm{d}x & &\textrm{(4)} \\ y&=e^{-F(x)} \int e^{F(x)}\cdot g(x)\,\mathrm{d}x & &\textrm{(5)} \end{align} $$


 * 1) The given differential equation.
 * 2) Multiplying with $$e^{F(x)}$$ on both sides, since $$F(x)$$ in continuer.
 * 3) Using the product rule for differentiation to simplify the left side.
 * 4) Integrating on both sides.
 * 5) Multiplying with $$e^{-F(x)}$$ on both sides.

It is to be noted, that the constant $$C $$ needed to define the particular solution is not missing but hidden as the constant of integration.