Bernoulli's EquationDate: 12/7/95 at 10:18:33 From: Anonymous Subject: Differential equation Dear Dr. Math, I'm trying to solve the following equation: y'=a*y-b*y*y*y, y(0)=y_0 I believe this is an Riccati-differential-equation, but I don't know how to solve it. Maybe you have a hint for me. Greetings from Germany, Andrea Date: 5/30/96 at 11:3:36 From: Doctor Anthony Subject: Re: Differential equation This is an example of Bernoulli's equation. We have dy/dx = ay - by^3 dy/dx - ay = -by^3 Divide through by y^3 (1/y^3)dy/dx) - a/y^2 = -b Make the substitution u = 1/y^2 So du/dx = (-2/y^3)(dy/dx) (-1/2)du/dx = (1/y^3)dy/dx) Substitute for y and dy/dx (-1/2)du/dx) - au = -b du/dx + 2au = 2b Multiply by the integrating factor e^(INT(2a.dx)) = e^(2ax) e^(2ax).du/dx + 2au.e^(2ax) = 2b.e^(2ax) d/dx{u.e^(2ax)} = 2b.e^(2ax) Integrate u.e^(2ax) = 2b.INT{e^(2ax).dx} u.e^(2ax) = 2b(1/(2a)).e^(2ax) + const. (1/y^2).e^(2ax) = (b/a).e^(2ax) + const -Doctor Anthony, The Math Forum |
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