How Bernoulli differential equation arise naturally?
Solution 1:
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...
$$ y(x) \rightarrow v(t) \quad P(x) \rightarrow \mu \quad Q(x) \rightarrow \nu $$
Solution 2:
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:- http://arxiv4.library.cornell.edu/PS_cache/arxiv/pdf/1011/1011.0060v1.pdf
Hope that helped!
Solution 3:
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
Solution 4:
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.
Solution 5:
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.