Bernoulli Differential Equation


Let v=y^(1-n) for n!=1. Then


Rewriting (1) gives


Plugging (4) into (3),


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


where P(x)=(1-n)p(x) and Q(x)=(1-n)q(x). It can therefore be solved analytically using an integrating factor


where C is a constant of integration. If n=1, then equation (◇) becomes


The general solution is then, with C_1 and C_2 constants,

 y={[((1-n)inte^((1-n)intp(x)dx)q(x)dx+C_1)/(e^((1-n)intp(x)dx))]^(1/(1-n))   for n!=1; C_2e^(int[q(x)-p(x)]dx)   for n=1.

Explore with Wolfram|Alpha


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.

Subject classifications