Let y0=y, y1= y', y2= y''. They y'''= y2' so the equation says y1'= sin(t)e^{y0}+ (y0)^2+ (y1)^{4/3} and y0'= y1. You can write those two equations as the single matrix equation

with initial condition

Now do you know the conditions for existence and uniqueness of solutions to a first order equation? Apply them to this equation.