Solving Non-linear First Order ODEs with Variable Coefficients?

[member 428835]

Homework Statement

$$y' y + \frac{y}{x} = 1 - 2x$$

Homework Equations

nothing comes to mind

The Attempt at a Solution

i've guessed a quadratic but that didn't work. now I'm stuck. any ideas? also, this is not homework, but a problem I am working on.

Thanks!

 

[BiGyElLoWhAt]

Is y y(x)?

 

[member 428835]

sorry, yes, $y = y(x)$

 

[LCKurtz]

Nonlinear equations can seem deceptively simple. This is a special case of an Abel DE of the second kind. This link may or may not be helpful:
http://eqworld.ipmnet.ru/en/solutions/ode/ode0125.pdf

 

[member 428835]

Nonlinear equations can seem deceptively simple. This is a special case of an Abel DE of the second kind. This link may or may not be helpful:
http://eqworld.ipmnet.ru/en/solutions/ode/ode0125.pdf

This is an interesting website. When I perform their substitutions I arrive at $$y \frac{dy}{dz} -y = \frac{2x - 1}{\ln | x |} : z:= \int -\frac{1}{x} dx$$ but from here their table provides no further help. Do you suppose this problem has never been analytically solved?

 

[LCKurtz]

This is an interesting website. When I perform their substitutions I arrive at $$y \frac{dy}{dz} -y = \frac{2x - 1}{\ln | x |} : z:= \int -\frac{1}{x} dx$$ but from here their table provides no further help. Do you suppose this problem has never been analytically solved?

I have no idea but it wouldn't surprise me.

 

[BiGyElLoWhAt]

This did appear deceptively easy at first glance =/