Solving Degree 2 Partial differential equation

Question

Please, can anyone help me solving this problem?

$$\frac{\partial U}{\partial t}−(\frac{\partial U}{\partial x})^2=0$$

where $U=U(x,t)$

with side condition as $U(x,0)=\cos x$

Answer 1

$$U_t-(U_x)^2=0$$ HINT : $$U_{xt}-2U_{xx}U_x=0$$ Let $\quad V(x,t)=U_x(x,t)$

$V_x=U_{xx}$ and $V_t=U_{xt}$ $$V_t-2V_xV=0$$ This is a Burger's equation which is well documented in the literature.

https://en.wikipedia.org/wiki/Burgers%27_equation