MHB How can nonlinear ODEs be solved effectively?

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To solve the nonlinear ODE presented, clarification on the equation's structure is essential, particularly regarding the placement of parentheses in the exponential function. A suggested substitution is to let \( u = \frac{x+y}{2x-y+1} \), which simplifies the equation to a separable form. The resulting equation leads to the expression \( \frac{e^{-u}}{u+1}\,u'=\frac{1}{3x+1} \). However, it is noted that the integral on the left side is not elementary, indicating potential challenges in finding a closed-form solution. Effective strategies for tackling nonlinear ODEs often involve substitutions and recognizing separability, even when integrals may not be straightforward.
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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!
 
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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.
 
If it is indeed what Ackbach says, try letting $u = \dfrac{x+y}{2x-y+1}$.
 
Jester said:
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.
 
Last edited:

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