## First order linear differential equation

1. The problem statement, all variables and given/known data
Solve this differential equation:
(y^2 +1)*dx + (2xy + 1)*dy = 0

2. Relevant equations
dy/dx + P(x)*y = Q(x)
u(x) = e^(integral of P(x)dx)
(d/dx)(u(x)*y) = Q(x)*u(x)
y = (integral of (Q(x)*u(x)dx))/(u(x)

3. The attempt at a solution
I tried dividing by dx then distributing and rearranging to get it into the right form, but run into problems:
y^2 + 1 + 2xy*dy/dx + dy/dx = 0
dy/dx + y/2x = (-1/(2xy))(dy/dx) -1/(2xy)

this is the closest I could get it to the right form. It would give me u(x) = x^(1/2), but I wouldn't be able to integrate the right side as it would have both x and y. Is there a way to get past this, or did I just rearrange poorly? I just integrated it anyway and the part that was integrated with respect to x I held y as constant, and vice versa, but I'm sure it's wrong so I won't show how I did that. This is for a calc II class so it should be doable without any advanced tricks.

thanks!
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 Recognitions: Homework Help Science Advisor That's an exact differential. You should show us how you did it wrongly by integrating. Because you should be able to do it that way.
 that's interesting. We haven't done exact differentials as far as I know, but since I don't know what that means perhaps we have! dy/dx + y/2x = (-1/(2xy))(dy/dx) -1/(2xy) using u(x) = x^(1/2) integral of ((d/dx)(y*x^(1/2))dx) = integral of (-1/(2*x^(1/2)*y)(dy/dx)(dx)) - integral of (-1/(2*x^(1/2)*y)(dx)) I integrated with respect to y for the first one on the right side, treating x as a constant, and integrated with respect to x on the right side, holding y constant, to get: x^(1/2)*y = -1/(2*x^(1/2))*ln(abs(y)) - (x^(1/2))/y + C y = -1/(2*x)*ln(abs(y)) - 1/y + C/(x^(1/2))

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## First order linear differential equation

Another way to about doing this is to consider what d/dx( xy2) works out to be i.e. d(xy2)

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