How Does the Constant Gradient Condition Affect Solutions to the Wave Equation?

[chaotixmonjuish]

Let u be a solution of the wave equation u_tt-u_xx=0 on the whole plane. Suppose that u_x(x,t) is a constant on the line x=1+t. Assume that u(x,0)=1 for all x in R and u(1,1,)=3. Find such a solution u.

I need help trying to incorporate the u_x(x,t) is a constant on the line x=1+t

 

[chaotixmonjuish]

So I got this as a solution by plugging it into an equation for wave equations with a Neumann condition:

u(x,t)= t+1/2(4t+2t^2) x>0

u(x,t)= t+1/2(3/2+5t+3t^2/2) 0<x<t