 |
(1)
|
令 
其中 
。 则
 |
(2)
|
重写方程 (1) 得到
 |  |  |
(3)
|
 |  |  |
(4)
|
![(dv)/(dx)=(1-n)[q(x)-vp(x)].](/images/equations/BernoulliDifferentialEquation/NumberedEquation3.svg) |
(5)
|
 |
(6)
|
其中 
和 
。 因此可以使用 积分因子 解析求解
 |  |  |
(7)
|
 |  |  |
(8)
|
其中 
是积分常数。 如果 
，则方程 (◇) 变为
 |
(9)
|
 |
(10)
|
![y=C_2e^(int[q(x)-p(x)]dx).](/images/equations/BernoulliDifferentialEquation/NumberedEquation7.svg) |
(11)
|
则通解是，其中 
和 
是常数，
![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.](/images/equations/BernoulliDifferentialEquation/NumberedEquation8.svg) |
(12)
|
伯努利微分方程
使用 Wolfram|Alpha 探索
参考文献
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.在 Wolfram|Alpha 中被引用
伯努利微分方程引用为
Weisstein, Eric W. "伯努利微分方程。" 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/BernoulliDifferentialEquation.html