What you have looks good! Once you got it wasn't really necessary to go any further. I agree with you, having independent of seems odd but it satisfies both the PDE and IC.
Solve the following Cauchy problem
The characteristic equations are , , .
The initial conditions are , and .
The Jacobian is and hence we expect a unique solution when and . (Is this correct?)
Now solve the characteristic equations.
Apply initial condition to get and hence .
Apply initial condition to get and hence . (Is this it for the question? Why is independent of ? What have I done wrong?)
Substitute above and into characteristic equation and we get . Integrate over and we get . Apply initial condition we get and .
From expressions of and obtained above we get
Therefore the characteristics is . (Do I need this characteristics at all? What should I do with it?)
Is the above attempt correct?