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September 8th, 2017, 04:20 AM   #1
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differential equations

Can someone please explain to me how to find differential equation of family of curves? Here is an example:

y=Ce^((x^2+y^2)/2)
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September 8th, 2017, 05:21 AM   #2
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y = cg(x)$\implies$y' = cg'(x)$\implies$g(x)y' = cg(x)g'(x) = yg'(x),
so the desired equation is g(x)y' - g'(x)y = 0.

Sometimes, eliminating c is more difficult.
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September 8th, 2017, 05:26 AM   #3
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Does y really occur in two places in that?

A differential equation involves derivatives so we start by differentiating!

$\displaystyle y(x)= Ce^{\frac{x^2+ y^2}{2}}$ so

$\displaystyle y'= 2Cxe^{\frac{x^2+ y^2}{2}}+ 2Cye^{\frac{x^2+ y^2}{2}}y'$
$\displaystyle y'= 2C\left(x+ yy'\right)e^{\frac{x^2+ y^2}{2}}$

To get an equation that gives that whole family of curves we need to eliminate "C" which we can do by dividing that by the original equation:

$\displaystyle \frac{y'}{y}= 2\left(x+ yy'\right)$

That answer can, of course, be rewritten in a number of different ways: perhaps as $\displaystyle y'= 2xy+ 2y^2y'$ and then $\displaystyle (1- 2y^2)y'= 2xy$ so that $\displaystyle y'= \frac{2xy}{1- 2y^2}$
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September 8th, 2017, 06:32 AM   #4
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Quote:
Originally Posted by Country Boy View Post
Does y really occur in two places in that?

A differential equation involves derivatives so we start by differentiating!

$\displaystyle y(x)= Ce^{\frac{x^2+ y^2}{2}}$ so

$\displaystyle y'= 2Cxe^{\frac{x^2+ y^2}{2}}+ 2Cye^{\frac{x^2+ y^2}{2}}y'$
$\displaystyle y'= 2C\left(x+ yy'\right)e^{\frac{x^2+ y^2}{2}}$

To get an equation that gives that whole family of curves we need to eliminate "C" which we can do by dividing that by the original equation:

$\displaystyle \frac{y'}{y}= 2\left(x+ yy'\right)$

That answer can, of course, be rewritten in a number of different ways: perhaps as $\displaystyle y'= 2xy+ 2y^2y'$ and then $\displaystyle (1- 2y^2)y'= 2xy$ so that $\displaystyle y'= \frac{2xy}{1- 2y^2}$
Their answer is : -1/y' = xy/(1-y^2)
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September 8th, 2017, 09:12 AM   #5
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Their answer is incorrect, and the answers that Country Boy gave are also incorrect.

When replying, please don't quote the whole previous post unnecessarily.
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