11.5.5.3 Mixed problems

Problem: Solve the mixed problem: \[ \left\{\begin{array}{c} u_{t t}=u_{x x}+6 t, \quad t>0, \quad x>0 \\ \left.u\right|_{t=0}=2 x,\left.\quad u_{t}\right|_{t=0}=0,\left.\quad u\right|_{\chi=0}=t^{3} \end{array}\right. \]