= Solution
Multiplying the <ordinary differential equation> by the nonzero <integrating factor> from part (i) makes its left-hand side $dF(x,y(x))/dx$. Hence the potential is constant:
$$
y\sqrt{16+x^2}-\frac13(16+x^2)^{3/2}=C.
$$
Since $16+x^2>0$ for every real $x$, this is equivalent to the explicit general solution
$$
\boxed{y(x)=\frac{16+x^2}{3}+\frac{C}{\sqrt{16+x^2}},\qquad C\in\mathbb R.}
$$
It is defined for all real $x$. There are no lost solutions, since the <integrating factor> and the coefficient of $y'$ are everywhere nonzero.
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