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Date: 30-12-2018
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Date: 26-12-2018
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Date: 30-5-2018
669
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(1) |
Let for . Then
(2) |
Rewriting (1) gives
(3) |
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(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) |
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(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.
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