MHB How can nonlinear ODEs be solved effectively?

[hatguy]

I need to solve the following ODE:

http://www.sosmath.com/CBB/latexrender/pictures/041ee1419e05bc0776451b294c1dcc0e.png

but i can't figure out a way to. Please help!

 

[Ackbach]

Regarding 3:
Did you mean
$$y'=e^{ \frac{x+y}{2x-y+1}}+\frac{3y-1}{3x+1}?$$
The absence of a closing parenthesis in the numerator of the argument of the exponential function makes your meaning unclear.

 

[Mathwebster]

If it is indeed what Ackbach says, try letting $u = \dfrac{x+y}{2x-y+1}$.

 

[Ackbach]

If it is indeed what Ackbach says, try letting $u = \dfrac{x+y}{2x-y+1}$.

Nice! The result is separable. I get
$$\frac{e^{-u}}{u+1}\,u'=\frac{1}{3x+1}.$$

Of course, the integral on the left is not elementary. Oh, well.