LinearEq

LinearEq - First-Order Linear O.D.E. and Bernoulli...

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First-Order Linear O.D.E. and Bernoulli Equations a- Linear Equations Definition : A first-order O.D.E. is linear in the dependent variable y and the independent variable x if it can be expressed as A(x) dy dx + B(x)y = C(x) Remark : If we divide each term by A(x), we get dy dx + P(x)y = Q(x), where P(x) = B(x) A(x) and Q(x) = C(x) A(x) Exampl e: The equation x dy dx + (x + 1)y = x 3 is linear where A(x) = x, B(x) = x + 1 and C(x) = x 3 . We can rewrite the equation in the form dy dx + (1 + 1 x )y = x 2 , where P(x) = 1 + 1 x and Q(x) = x 2 Let’s rewrite dy dx + P(x)y = Q(x) in differential form [P(x)y – Q(x)] dx + dy = 0 where M(x,y) = P(x)y – Q(x) and N(x,y) = 1 since ! M(x,y) ! y = P(x) and ! N(x,y) ! x = 0 , the equation is not exact unless P(x) = 0. Let’s find an integrating factor. Assume that the integrating factor depends on x, only, let’s call it n(x). Then, multiplying the equation by n(x), we get n(x)[P(x)y – Q(x)] dx + n(x) dy = 0 and it is an exact equation, Therefore, ! ! y n(x)P(x)y " n(x)Q(x) [ ] = n(x)P(x) = ! ! x n(x) [ ] or n(x)P(x) = dn dx , where P(x) is a known function of x, but n(x) is an unknown function. Solving the differential equation we can determine n(x).
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This note was uploaded on 07/23/2011 for the course MAP 2302 taught by Professor Staff during the Fall '08 term at FIU.

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LinearEq - First-Order Linear O.D.E. and Bernoulli...

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