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I came across a highly nonlinear 4th order ODE, with multiple derivative product terms, after applying a transformation to an even more complex ODE. I have two queries, one rather specific and other more general.

  1. Is there a general solution to $A^2 \nu G^{(4)}(\eta )-4 A \nu G^{(3)}(\eta )-A G(\eta ) G^{(3)}(\eta )+A G'(\eta ) G''(\eta )+4 \nu G''(\eta )+2 G(\eta ) G''(\eta )=0$

$A$ and $\nu$ are constants from the transformation which for our purposes is arbitrary. It does have a trivial solution $G(\eta) = \alpha \eta + \beta$, the linear function. I could not find any other solutions with all the usual methods. Can somebody suggest a different solution or a method to solve the equation?

  1. How do we approach such ODE's with multiple derivative product terms? i.e the kind of ODE's which are nonlinear but without any other functions appearing in it. I mean ODE's of the type $L\left(G^{(4)}(\eta),G^{(3)}(\eta),G^{(2)}(\eta),G^{(1)}(\eta),G(\eta)\right)=0$
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No, there is no general solution to this equation or similar ones. There are a few things we can try, however, in some specific cases.

Since first order equations are generally easier to solve, we can turn a second-order autonomous equation into a first order non-autonomous one by making a substitution $F(G(\eta)) = G'(\eta)$, which gives an equation for $F(G)$. You can use this same trick for higher-order equations to reduce the order by one at the cost of losing autonomy. Generally, this is not a good trade-off, but you may get lucky.

The next thing is to write the equation as a first order system by making substitutions $x_1 = G, x_2 = G', \dots$ and so on, and writing the system $X' = A(X)$. The advantage of this is that you can more easily identify fixed points and study stability and find approximate solutions nearby. The downside, however, is that it's not getting you closer to an analytic solution.

A "happy-medium" between the two might be to use the nonlinear equivalent of power series solutions, which is perturbation method. For example, multiple scales entails writing $G = \sum_{i=1}^\infty \epsilon^iG_i$, $\eta = \sum_{i=1}^\infty \epsilon^i\eta_i$, and following through with chain rule to change the single equation into a series of equations for the $G_i$. The upside here is that (under certain conditions) this approximation will converge to a solution, and this expansion can be used to analyze how the solutions change with respect to changes in the parameters ($A,\nu$ here), known as bifurcations. The downsides are that is requires you to change the equations in some cases where the nonlinearities are not already "order $\epsilon$" by crudely putting an $\epsilon$ in front of any nonlinear terms, and removing it later. Also, computational complexity goes up quite a bit for each equation in the series.

The next best thing is, of course, numerical methods. They will give you accurate approximations of the value of the solution, as a series of points. The downside is that you will get nothing analytical to work with in the end, so if you need a formula you can write down, this won't work.

Overall, none of these methods are very satisfactory in general, and each one gives you different information in different cases -- this is simply the state of nonlinear ordinary differential equations right now. I hope this sheds some light on your issue.

Lastly, though it doesn't help in this case, you may want to bookmark http://eqworld.ipmnet.ru/index.htm in case you find yourself looking for analytic solutions to nonlinear equations frequently.

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