My Math Forum Separation of Variables

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 February 21st, 2011, 11:06 PM #1 Member   Joined: Jul 2010 Posts: 34 Thanks: 0 Separation of Variables I'm trying to solve the following ODE - (2xy -5)dx + (x2 + y2 )dy = 0, initial conditions y(3) =1; but I can't even separate the variable successfully. My workings so far - (2xy -5)dx + (x^2 + y^2 )dy = 0 2xydx - 5dx + x^2dy + y^2dy = 0 2xydx + x^2dy + y^2dy = 5dx 2xy + (x^2dy)/dx +( y^2dy)/dx = 5 2xy + X^2 y' + y^2 y' = 5 (SUBSTITUTING dy/dx for y') 2xy/y' + x^2 + y^2 = 5/y' Now, if I'm on the right track (which I suspect I'm not) I just need to separate the that first term - 2xy/y'.. I think. Help.
 February 21st, 2011, 11:57 PM #2 Senior Member   Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3 Re: Separation of Variables It's exact. Integrating the dx part with respect to x, and the dy part with respect to y, you get x²y - 5x + f(y) and x²y + y³/3 + g(x) respectively, so you have $\int \mathrm{d}(x^2y-5x+\frac13y^3)= \int 0\\\therefore\quad x^2y-5x+\frac13y^3 = C$
 February 22nd, 2011, 02:06 PM #3 Member   Joined: Jul 2010 Posts: 34 Thanks: 0 Re: Separation of Variables I see. I was unaware of exact DEs. Thanks for the solution and the link.

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