Bernoulli differential equation

:"This topic in mathematics is named after Jakob Bernoulli. See Bernoulli's principle for an unrelated topic in fluid dynamics, named after the inventor Daniel Bernoulli."

In mathematics, an ordinary differential equation of the form

:y'+ P(x)y = Q(x)y^n,

is called a Bernoulli differential equation or Bernoulli equation when n≠1, 0. Dividing by y^n yields:frac{y'}{y^{n + frac{P(x)}{y^{n-1 = Q(x).A change of variables is made to transform into a linear first-order differential equation.:w=frac{1}{y^{n-1:w'=frac{(1-n)}{y^{ny':frac{w'}{1-n} + P(x)w = Q(x)

The substituted equation can be solved using the integrating factor

:M(x)= e^{(1-n)int P(x)dx}.


Consider the Bernoulli equation :y' - frac{2y}{x} = -x^2y^2Division by y^2 yields:y'y^{-2} - frac{2}{x}y^{-1} = -x^2Changing variables gives the equations:w = frac{1}{y}:w' = frac{-y'}{y^2}.:w' + frac{2}{x}w = x^2which can be solved using the integrating factor:M(x)= e^{2int frac{1}{x}dx} = x^2.Multiplying by M(x),:w'x^2 + 2xw = x^4,, Note that left side is the derivative of wx^2. Integrating both sides results in the equations:int (wx^2)' dx = int x^4 dx:wx^2 = frac{1}{5}x^5 + C :frac{1}{y}x^2 = frac{1}{5}x^5 + CThe solution for y is:y = frac{x^2}{frac{1}{5}x^5 + C}

External links


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Ordinary differential equation — In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable. A simple example is Newton s second law of… …   Wikipedia

  • Euler-Bernoulli beam equation — Euler Bernoulli beam theory, or just beam theory, is a simplification of the linear theory of elasticity which provides a means of calculating the load carrying and deflection characteristics of beams. It was first enunciated circa 1750, but was… …   Wikipedia

  • Dispersive partial differential equation — In mathematics, a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive. In this context, dispersion means that waves of different wavelength propagate at different phase velocities.… …   Wikipedia

  • Bernoulli equation — may refer to:*Bernoulli differential equation *Bernoulli s equation, in fluid dynamics. *Euler Bernoulli beam equation, in solid mechanics …   Wikipedia

  • Bernoulli — can refer to: *any one or more of the Bernoulli family of Swiss mathematicians in the eighteenth century, including: ** Daniel Bernoulli (1700–1782), developer of Bernoulli s principle ** Jakob Bernoulli (1654–1705), also known as Jean or Jacques …   Wikipedia

  • Bernoulli's principle — This article is about Bernoulli s principle and Bernoulli s equation in fluid dynamics. For Bernoulli s Theorem (probability), see Law of large numbers. For an unrelated topic in ordinary differential equations, see Bernoulli differential… …   Wikipedia

  • Bernoulli family — The Bernoullis were a family of traders and scholars from Basel, Switzerland. The founder of the family, Leon Bernoulli, immigrated to Basel from Antwerp in the Flanders in the 16th century.The Bernoulli family has produced many notable artists… …   Wikipedia

  • bernoulli equation — noun see bernoulli s equation * * * 1. Hydrodynamics. See Bernoulli s theorem (def. 2). 2. Also called Bernoulli s differential equation. Math. a differential equation of the form dy + f(x)y = g(x …   Useful english dictionary

  • Bernoulli equation — 1. Hydrodynamics. See Bernoulli s theorem (def. 2). 2. Also called Bernoulli s differential equation. Math. a differential equation of the form dy + f(x)y = g(x)yn, where n is any number other than 0 or 1. [1915 20; (in def. 1) named after Daniel …   Universalium

  • Bernoulli family — Two generations of distinguished Swiss mathematicians. Jakob (1655–1705) and Johann (1667–1748) were the sons of a pharmacist who wanted one boy to study theology and the other medicine. Over his objections, both pursued careers in mathematics,… …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.