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# Bernoulli Differential Equation

 (1)

Let for . Then

 (2)

Rewriting (1) gives

 (3) (4)

Plugging (4) into (3),

 (5)

Now, this is a linear first-order ordinary differential equation of the form

 (6)

where and . It can therefore be solved analytically using an integrating factor

 (7) (8)

where is a constant of integration. If , then equation (◇) becomes

 (9)
 (10)
 (11)

The general solution is then, with and constants,

 (12)

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## References

Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 5th ed. New York: Wiley, p. 28, 1992.Ince, E. L. Ordinary Differential Equations. New York: Dover, p. 22, 1956.Rainville, E. D. and Bedient, P. E. Elementary Differential Equations. New York: Macmillian, pp. 69-71, 1964.Simmons, G. F. Differential Equations, With Applications and Historical Notes. New York: McGraw-Hill, p. 49, 1972.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 413, 1995.Zwillinger, D. "Bernoulli Equation." §II.A.37 in Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 120 and 157-158, 1997.

## Referenced on Wolfram|Alpha

Bernoulli Differential Equation

## Cite this as:

Weisstein, Eric W. "Bernoulli Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BernoulliDifferentialEquation.html