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\begin{align*} u_{tt}-c^2u_{xx}=0, x>0\\ u(x,0)=u_t(x,0)=0\\ u_x(0,t)=\frac{t}{1+t^2},t>0 \end{align*}

According to the textbook, I should look for solutions in the form $u(x,t)=F(x-ct)$ and impose the boundary conditions to get an ODE. However, I do not know how to impose the initial condition after these steps. Thanks!

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