BERNOULLI�s DIFFERENTIAL EQUATIONS - Copy (2)

BERNOULLI�s DIFFERENTIAL EQUATIONS - Copy (2) - A....

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Unformatted text preview: A. Alaca MATH 1005 Winter 2010 2 BERNOULLI’s DIFFERENTIAL EQUATIONS Definition: An equation of the form y + P (x)y = Q(x) y n , where n is any real number is called Bernouli’s equation. Note: • When n = 0, we have a first order lin. diff. eqn. • If n = 1 (and y = 0), then y + P (x) = Q(x) y y = Q(x) − P (x) y d (ln y ) = Q(x) − P (x) dx ln y = • Let n = 0, 1. Then the substitution u = y 1− n transforms Bernoulli’s equation into a first order linear equation. du du dy dy = = (1 − n)y −n dx dy dx dx y n du dy = . dx 1 − n dx Then, y + P (x)y = Q(x) y n becomes: 1 du + P (x)y = Q(x)y n yn 1 − n dx 1 du + P (x)y 1−n = Q(x) 1 − n dx du + (1 − n)P (x)y 1−n = (1 − n)Q(x) dx du + (1 − n)P (x)u = (1 − n)Q(x) , dx which is a first order lin. diff. eqn. (Q(x) − P (x)) dx ...
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This note was uploaded on 03/08/2010 for the course MATH 1005 taught by Professor Any during the Winter '07 term at Carleton CA.

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