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A Bernoulli differential equation is a non-linear differential equation of the form $$ \frac{dy}{dx} + P(x)y = Q(x)y^n. $$

I understand this is special; Because its exact solution is known though it's non-linear (in other words, a substitution $w = y^{1 - n}$ makes the equation linear).

However, how does this equation arise naturally? Is there any physical meaning in specific case? If so how to derive this equation?

Thank you.

Edit: I'm interested in how this equation naturally arise rather than its historical origin. I also appreciate if you give me a natural interpretation of this equation. I don't care whether its background is physics, chemistry, or geometry (though I hope the example is elementary enough so that I can understand).

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It's a generalization of the frictional forces equations.

  • The ideal case:

$$ a = \dot{v} = -\mu\,v $$

Exponential solution. Particle don't stop in finite time.

  • More realistic case:

$$ \dot{v} = -\mu\,v + \nu\,v^3 $$

Particle stop in finite time.

Odd exponents because of friction opposes movement.

  • Why drag this equation?

The frictional force is a function of movement $F(v)$. It has the property $F(-v) = -F(v)$ (friction opposes movement). Then the Taylor series:

$$ F(v) \approx a_1\,v + a_3\,v^3 + a_5\,v^5 + \dots $$

$F(v)$ (accordingly $a_i$'s) depends on the physical system: floor, air...

$$ y(x) \rightarrow v(t) \quad P(x) \rightarrow \mu \quad Q(x) \rightarrow \nu $$

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    $\begingroup$ Thank you for your answer! It seems that this is what I wanted. But let me ask a few questions. A concrete example of the ideal case is, for example, throw a ball in pool, right? The horizontal velocity is determined by Stokes's law: $m\dot{v} = -6\pi \mu r v$. I understand why more realistic case takes the form from the argument of taylor expansion, but I've never seen such example. Would you give me a specific example? $\endgroup$
    – Orat
    Commented Apr 6, 2013 at 2:22
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    $\begingroup$ $m\dot{v} = -6\pi \mu r v$ is correct but is ideal, only for slow motion, linear Reynolds regime. When the speed increases we need more terms of Taylor series. For example nonlinear Reynolds regime. $\endgroup$
    – Edoot
    Commented Apr 6, 2013 at 3:19
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    $\begingroup$ For references see 1 and 2 $\endgroup$
    – Edoot
    Commented Apr 6, 2013 at 3:35
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    $\begingroup$ A minor question: do you have any reason that you write $\dot{v} = -\mu v + \nu v^3$ instead of $\dot{v} = -\mu v - \nu v^3$? $\endgroup$
    – Orat
    Commented Apr 6, 2013 at 3:55
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    $\begingroup$ a1 is <0 but a3 sometimes <0 and sometimes >0. $\endgroup$
    – Edoot
    Commented Apr 6, 2013 at 9:06
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As far as I know, there was no particular motivation behind the development of the Bernoulli equation. The Bernoulli family was quite renowned for taking up hard challenges in mathematics, and giving solutions for particular cases, or sometimes brilliant generalizations. In other words, they often solved problems for the challenge & thrill, rather than for potential applications. As for its derivation, Jakob Bernoulli must have probably just seen it as a natural extension to the then-existing theory of differential equations, and thus worked on solving it.

Having said that though, modern physics indeed uses Bernoulli differential equations for modelling the dynamics behind certain circuit elements, known as Bernoulli memristors. I do not know much about the details, but if you're curious, this paper might be of interest to you:- P. S. Georgiou, S. Yaliraki, M. Barahona, E. M. Drakakis. Quantitative Measure of Hysteresis for Bernoulli Memristors. 2010. arXiv:1011.0060v1

Hope that helped!

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Perhaps this brief history of differential equations will shed some light on the issue for you: http://www.math.ou.edu/~mleite/MATH3413_sp11pdf/ODE_History.pdf

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    $\begingroup$ Thank you for your answer! But the only sentence I found that relevant to this question is "The equation: $dy/dx + P(x)y = f(x)y^n$ known today as the Bernoulli equation, was proposed for solution by James Bernoulli in December, 1695 [3]. The following year Leibniz solved the equation by making substitutions and simplifying to a linear equation, similar to the method employed today [18]." Is there any other information that I miss? $\endgroup$
    – Orat
    Commented Mar 28, 2013 at 17:13
  • $\begingroup$ The link to the PDF is broken, but a snapshot is saved on the Wayback Machine. $\endgroup$
    – The Amplitwist
    Commented Jun 19, 2022 at 16:33
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The Bernoulli differential equation also show up in some economic utility maximization problems. For an example, see Robert Merton's paper Lifetime Portfolio Selection under Uncertainty (1969). Equation 23 is the Bernoulli diff. eqn. subject to a boundary condition. It is not easy to interpret in context here but none the less it is an equation that shows up.

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If you model the movement of a body under Newton law, and you want to model the friction as well, you add to the second order differential equation forces related to the velocity of the body.

Models of frictions may require dependency on the power of the velocity.

Now, in order to solve the second order differential equation you may opt to trasform it into a system of first order differential equation. If you do so, then you get a Bernoulli equation as the "important" equation in the system.

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  • $\begingroup$ Thank you for your answer! I guess what you have in your mind is the motion of (big) rain drop described by $m\dot{v} = mg - kv^2$, right? But it looks not the case. Would you give me a concrete example? $\endgroup$
    – Orat
    Commented Apr 2, 2013 at 19:30
  • $\begingroup$ I come up with realistic example. Thank you. See my comment under the answer of Edoot. $\endgroup$
    – Orat
    Commented Apr 6, 2013 at 2:23

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