 |
(1)
|
Let 
for 
.
Then
 |
(2)
|
Rewriting (1) gives
Plugging (4) into (3),
![(dv)/(dx)=(1-n)[q(x)-vp(x)].](/images/equations/BernoulliDifferentialEquation/NumberedEquation3.svg) |
(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
where 
is a constant of integration. If 
, then equation (◇) becomes
 |
(9)
|
 |
(10)
|
![y=C_2e^(int[q(x)-p(x)]dx).](/images/equations/BernoulliDifferentialEquation/NumberedEquation7.svg) |
(11)
|
The general solution is then, with 
and 
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.](/images/equations/BernoulliDifferentialEquation/NumberedEquation8.svg) |
(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
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