= Solution
For a classical solution take $F,G$ twice differentiable, so that the second <derivatives> exist. Direct use of the <chain rule> gives
$$
y_{tt}=25(F''+G''),\qquad y_{xx}=4(F''+G''),
$$
with the two profile arguments as specified. Thus $4y_{tt}=25y_{xx}$.
To prove generality, introduce <characteristic coordinates> $\xi=2x+5t$, $\eta=2x-5t$, whose Jacobian <determinant> is $-20\ne0$. For $y=U(\xi,\eta)$ the differential operator is
$$
4\partial_t^2-25\partial_x^2=-400\partial_\xi\partial_\eta.
$$
The equation becomes $U_{\xi\eta}=0$. Integrating once in each variable yields
$$
\boxed{U(\xi,\eta)=F(\xi)+G(\eta).}
$$
This is the <countertravelling decomposition of the one-dimensional wave equation>, with <wave speed> $5/2$. Once differentiability alone is not sufficient for a classical second-order equation; the displayed argument uses the required second <derivatives>.
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