= Solution
For a short string element $dx$, small slope makes its tension magnitude approximately constant and its vertical component approximately $Ty_x$. The net vertical force is $T[y_x(x+dx,t)-y_x(x,t)]\simeq Ty_{xx}dx$. Its <mass> is $\rho\,dx$. <Newton's second law> gives
$$
\rho\,dx\,y_{tt}=Ty_{xx}\,dx,\qquad \boxed{y_{tt}-c^2y_{xx}=0,\quad c=\sqrt{T/\rho}.}
$$
This assumes uniform equilibrium tension, small slopes, negligible bending stiffness and dissipation. The symbol $\rho$ is linear mass density here, not volume density.
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