Armour formular

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.[1]

The armour 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 ${\frac {\mathrm {d} y}{\mathrm {d} x}}+f(x)y=g(x)$ .[1]

Proof

{\begin{aligned}{\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{aligned}}

The given differential equation.
Multiplying with $e^{F(x)}$ on both sides, since $F(x)$ is continuous.
Using the product rule for differentiation to simplify the left side.
Integrating on both sides.
Multiplying with $e^{-F(x)}$ on both sides.[2]

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.[1]

References

Matematik for biovidenskab. Thomas Vils Pedersen, Henrik Laurberg Pedersen (2. udgave ed.). Frederiksberg: Biofolia. 2016. pp. 286–295. ISBN 978-87-91319-82-2. OCLC 954683829.CS1 maint: others (link)
"y' + h(x)y = g(x)". Matematisk bevissamling (in Danish). Retrieved 2021-11-13.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[:0-1] Matematik for biovidenskab. Thomas Vils Pedersen, Henrik Laurberg Pedersen (2. udgave ed.). Frederiksberg: Biofolia. 2016. pp. 286–295. ISBN 978-87-91319-82-2. OCLC 954683829.CS1 maint: others (link)

[2] "y' + h(x)y = g(x)". Matematisk bevissamling (in Danish). Retrieved 2021-11-13.

Armour formular

Proof

See also

References