Problem: Please solve the non-linear partial differential equation $\frac{\partial^2 u}{\partial t^2}+ \frac{\partial^2 u}{\partial x \partial t}= \left( \frac{\partial u}{\partial t} \right)^2+ \frac{\partial u}{\partial x} \cdot \frac{\partial u}{\partial t}$ with initial conditions $u(x, 0) = 0, \quad u_t(x, 0) = x^2$ . Please place your final solution in a $\boxed{}$ LaTeX Environment.
Solution: $\boxed{u(x, t) = -\ln\left(1 - \frac{t^3}{3} + x^2 t - x t^2\right)}$