I have been trying to solve this equation during fortnight $$ {u_{tt}}^2u_{ttxx} = 1. $$ But I still here. The only thing is change of variables $u_{tt}(t,x) = y(t,x) $ and solved the ODE $y'' = \frac{1}{y^2}$. But the solution $y(t,x)$ too complicated. I know that there are no common methods for solving such equation. But I wonder if somebody have any experience with this kind.
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A generic linear function in $t$ as $$u(x,t)=f_1(x)t+f_2(x)$$ does the job but, for the sake of completeness, I give here the result of Mathematica that involves ${\rm erf}^{1}$, the inverse of the error function, $$u(x,t)=f_1(x)t+f_2(x)+\int_1^tdt'\int_1^{t'}dt''e^{{\rm erf}^{1}\left[\frac{2}{\pi }\left(e^{C_1t''} (x+C_2t'')^2\right)\right]\frac{1}{2} C_1t''}.$$ 

