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)

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.