Classify general expressions for characteristic coordinates

Question

Solve the following problems:

a)  Classify and find general expressions for the characteristic coordinates for the equation

         u_xt+tu_tt+sin(x+1)=0

b)  Use the canonical coordinates  ξ =e^-xt and η=x and transfer the above PDE into the new coordinates.  Solve it in the new coordinates and show that

        u(x.t) =e^x∫^x₀e^-x cos(α+te^-x)dα+e^xF(te^-x)+G(x)  

where F and G are arbitrary functions of their arguments.