(a) Expand into powers of \(x\): \[\frac{dy}{dx}=x^{3/2}-6x^{1/2}+9x^{-1/2}.\] Integrating, \[y=\frac25x^{5/2}-4x^{3/2}+18x^{1/2}+c.\] At \((4,7)\), \[7=\frac25(32)-4(8)+18(2)+c=\frac{84}{5}+c,\] so \(c=-49/5\). Hence \[\boxed{y=\frac25x^{5/2}-4x^{3/2}+18x^{1/2}-\frac{49}{5}}.\]

(b) The curves differ only in their constant term, by \(k-7\). The transformation is a \(\boxed{\text{translation by }\begin{pmatrix}0\\k-7\end{pmatrix}}\).