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This is essentially a follow-up to When is a non-linear first-order ODE equivalent to a linear second-order ODE? The only case where a non-linear first-order ODE is equivalent to a linear second-order ODE is the Riccati equation, as explained in the answer to that question by Alexandre Eremenko.

On the other hand, there are certainly cases where a non-linear ODE of higher order can be made linear by some suitable transformation. For example, the non-linear second-order ODE $y(x)y''(x)-y'(x)^2=0$ becomes the linear ODE $u''(x)=0$ through $y(x)=e^{u(x)}$, and thus has solutions $y(x)=e^{c_1x+c_0}$.

What is known about more general cases? Is there a systematic way of telling whether a non-linear ODE such as $f(x)^3 f^{(4)}(x)-5 f(x)^2 f''(x)^2+5 f(x) f'(x)^2 f''(x)-f'(x)^4=0$ is equivalent to a linear ODE?

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    $\begingroup$ For the second-order case look at "λ-symmetry criteria for linearization of second order ODEs via point transformations" by Al-Dweik and colleagues (on Arxiv). An excerpt from the introduction: "It was shown by Lie that any second-order ODE $y''=f(x,y,y')$ which is linearizable via point transformations is at most cubic in the first derivative, i.e. it has the form \begin{align} y''+F_3(x,y)y'^3+F_2(x,y)y'^2+F_1(x,y)y'+F_0(x,y)=0.\end{align}" $\endgroup$ Commented Jan 31 at 18:49
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    $\begingroup$ For the second order case Tresse gives the conditions on the functions $F_j$ for which a transformation does exist, (and is quoted in Al-Dweik's paper). For the third-order case I do not know of a nice condition but you can look at point transformations of linear equations which all yield the form: \begin{align} y'''_{xxx}+[\Gamma_1 y'_x+\Gamma_0](y''_{xx})^2+[\Omega_3(y'_x)^3+\Omega_2(y'_x)^2+\Omega_1y'_x+\Omega_0]y''_{xx}+\Lambda_4 (y'_x)^4+\Lambda_3(y'_x)^3+\Lambda_2(y'_x)^2+\Lambda_1 y'_x+\Lambda_0=0, \end{align} for functions $\Gamma_j(x,y)$, $\Omega_j(x,y)$, $\Lambda_j(x,y)$. $\endgroup$ Commented Jan 31 at 18:59
  • $\begingroup$ @EliBartlett thanks, the suggestion to look at point transformations of linear equations to see what forms of non-linear equation can come up sounds very helpful! $\endgroup$ Commented Jan 31 at 20:02
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    $\begingroup$ Look up the paper: 'Reduction of fourth order ordinary differential equations to second and third order Lie linearizable forms' by Dutt and Qadir. In it they reference conditions for linearizablity of third-order equations and extend to certain cases of fourth order equations. $\endgroup$ Commented Feb 18 at 14:31

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