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Yes, you are right. I just wanted to mention that the related question, where the three mentioned curves have positive ranks, should be relatively easy to answer using the method from my paper.
You can refer to my paper, M. Ulas, "A Note on Higher Twists of Elliptic Curves," Glasg. Math. J. 52 (2010), no. 2, 371–381, where the question of the existence of simultaneous cubic/sextic twists of elliptic curves with $j=0$ with positive ranks is considered.
If $f(x)=2f_{1}(x)+1, g(x)=2g_{1}(x)+1$, where $f_{1}, g_{1}$ are given non-zero polynomials, then the polynomials $f(x)^{2^{n}}+g^{2^{n}}, f(x)^{2^{n}}+1$ as divisible by 2, are reducible in $\mathbb{Z}[x]$.
The answer is simple: you need to solve the last differential equation with repsect to $\phi$. This is an ordinary differential equation in one variable, so you have a lot of methods which can be used. Accorindg to Mathematica program the general solutions is: $\phi(y)=-\log \left(-\frac{2 \tanh ^{-1}\left(\frac{\tan \left(\frac{y}{2}\right) (\phi (y)(0)+1)}{\sqrt{1-(\phi (y)(0))^2}}\right)}{\sqrt{1-(\phi (y)(0))^2}}-c\right)$, where $c$ is a constant.
