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I have data (x, y) which I believe is generated by a differential equation of the following form:

$$A\frac{d^2y}{dx^2}+B(\frac{dy}{dx})^{C}+D=0$$

I can estimate the initial values, so given $A$, $B$, $C$, and $D$, I can numerically integrate this with RK4 very easily, but in my case I'm trying to regress the coefficients given a dataset (x, y). The solutions to this question almost meet my needs, but I'm hitting difficulties here due to the exponent on the first derivative.

I would strongly prefer to avoid blackbox methods like L-BFGS, but I can employ exhaustive search over one of the parameters (such as the exponent $C$), if knowing that parameter would simplify the rest of the problem. Any ideas, perhaps a convenient u-substitution?

Bananacus
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  • Could you joint to the question a numerical example Data(x,y) in order to check if a similar method as used in https://math.stackexchange.com/questions/3353495/ is applicable to your problem. – JJacquelin Feb 05 '24 at 09:06
  • It may be helpful to reduce the ODE to a parametric equation. Denoting $y'_x=p(x)$ and integrating gives \begin{align} x=k_1-A\int\frac{\mathrm dp}{Bp^C+D}. \end{align} Now take the Legendre transform $x=P$, $y=XP-Y$, $p=X$ to get \begin{align} P=k_1-A\int\frac{\mathrm dX}{BX^C+D}\longrightarrow Y=-k_0+k_1X-A\iint \frac{\mathrm dX\mathrm dX}{BX^C+D}. \end{align} If we take $X$ to be our parameter we have the parametric solution \begin{align} x=k_1-A\int\frac{\mathrm dX}{BX^C+D}, \quad y=k_0+A\iint\frac{\mathrm dX\mathrm dX}{BX^C+D}-AX\int \frac{\mathrm dX}{BX^C+D}. \end{align} – Eli Bartlett Feb 05 '24 at 23:56

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