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February 21st, 2011, 11:06 PM   #1
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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.
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February 21st, 2011, 11:57 PM   #2
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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 xy - 5x + f(y) and xy + y/3 + g(x) respectively, so you have

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February 22nd, 2011, 02:06 PM   #3
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Re: Separation of Variables

I see. I was unaware of exact DEs. Thanks for the solution and the link.
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