Solve $u_{tt}=7u_{xx},-\infty< x <\infty$ $u(x,0)=x^2, u_t(x,0)=\cos(3x),-\infty< x <\infty$
If you use d'Alembert's solution for this problem after doing change of variables and everything to get the solution $u(x,t)=F(x-ct)+G(x+ct)$ do you use the conditions $u(x,0)=x^2, u_t(x,0)=\cos(3x)$ for F and G? Example would the solution be $u(x,t)=(x-\sqrt{7}t)^2+\cos(3(x+\sqrt{7}t))$? If this is not the case where do we use the conditions?