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I was working on a physics project and have to solve this nonlinear ODE:-

$\dfrac{dy}{dx}=\dfrac{ky^3 +ey}{axy(y+b) + c(y-b)/\sqrt{x}}$,

where $a, b, c, k$ and $e$ are constants.

I haven't had much experience solving nonlinear ODEs, but I did try obtaining a Taylor series solution. It turned out to be far too complicated though. I do know there are simpler numerical methods, but I'm looking for an analytical solution. Can someone help me with this?

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  • $\begingroup$ Do you really need to solve it or qualitative behaviour is fine too? $\endgroup$ – Evgeny Feb 14 '14 at 17:51
  • $\begingroup$ Are you sure that an analytical solution exists? Unfortunately many nonlinear ODEs can not be solved analytically. $\endgroup$ – Doubt Feb 14 '14 at 17:52
  • $\begingroup$ @Doubt: I'm not familiar with uniqueness and existence theorems yet, so I can't say for sure. And I'm not aware of any other techniques that would tell me that, either. $\endgroup$ – Train Heartnet Feb 14 '14 at 18:09
  • $\begingroup$ @Evgeny: I do need to solve it, as the function obtained has to be used elsewhere. Behavior of the solution won't suffice. However, if such a solution does not exist, that would be valuable information. $\endgroup$ – Train Heartnet Feb 14 '14 at 18:12
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For the simple case where $a, b,c,k ,e = 1$, Mathematica gives the solution implicitly as \begin{align} \int_1^{y(x)}\frac{\xi-2}{\xi\left(1+\xi^2\right)^{7/4}\exp\left(\frac{3}{2}\arctan\xi\right)}\,d\xi = \frac{2x^{3/2}}{3\left(1+y^2(x)\right)^{3/4}\exp\left(\frac{3}{2}\arctan y(x)\right)} + c \end{align} There does not seem to be a closed-form expression for the integral, unfortunately. Probably this form is no more helpful than the original ODE in terms of solving the problem numerically.

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  • $\begingroup$ Thank you for trying! I guess this is the closest one could get to a solution. But just out of curiosity (and desperateness), if I had an initial condition like $y(0)=0$, would that help in any way? $\endgroup$ – Train Heartnet Feb 14 '14 at 19:53
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    $\begingroup$ No, the initial condition is not likely to make things easier. In fact, the initial condition $y(0) = 0$ may not be physically possible since $dy/dx|_{x=0,\,y=0}$ is undefined. Easier would be $y(0) = 1$, which gives $c = 0$. If $y(0) = 0$ should be physically possible, perhaps revisit the derivation of the ODE. $\endgroup$ – Doubt Feb 14 '14 at 20:29
  • $\begingroup$ On further reflection, I realized that neither $y(0)=1$, nor $y(0)=0$ are physically possible. The initial conditions are way more complicated than that. Thank you so much for all the help you've done though. I am accepting your answer, as that's the closest in terms of a solution. :) $\endgroup$ – Train Heartnet Feb 16 '14 at 4:57

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