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In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form ′ + = (), where is a real number.Some authors allow any real , [1] [2] whereas others require that not be 0 or 1.
Logistic differential equation (sometimes known as the Verhulst model) 2 = (()) Special case of the Bernoulli differential equation; many applications including in population dynamics [16] Lorenz attractor: 1
The logistic equation is a special case of the Bernoulli differential equation and has the following solution: f ( x ) = e x e x + C . {\displaystyle f(x)={\frac {e^{x}}{e^{x}+C}}.} Choosing the constant of integration C = 1 {\displaystyle C=1} gives the other well known form of the definition of the logistic curve:
Bernoulli equation may refer to: Bernoulli differential equation; Bernoulli's equation, in fluid dynamics; Euler–Bernoulli beam equation, in solid mechanics
Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert, and Euler. A simple example is Newton's second law of motion—the relationship between the displacement x {\displaystyle x} and the time t {\displaystyle t} of an object under the ...
In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form y ′ ( x ) = q 0 ( x ) + q 1 ( x ) y ( x ) + q 2 ( x ) y 2 ( x ) {\displaystyle y'(x)=q_{0}(x)+q_{1}(x)\,y(x)+q_{2}(x)\,y^{2}(x)} where q 0 ( x ...
The most celebrated of these is perhaps the Bernoulli flow. The Ornstein isomorphism theorem states that, for any given entropy H, there exists a flow φ(x, t), called the Bernoulli flow, such that the flow at time t = 1, i.e. φ(x, 1), is a Bernoulli shift. Furthermore, this flow is unique, up to a constant rescaling of time.
Jacob Bernoulli's paper of 1690 is important for the history of calculus, since the term integral appears for the first time with its integration meaning. In 1696, Bernoulli solved the equation, now called the Bernoulli differential equation, ′ = + ().