解：设大圆盘的直径为$ d_1\ \mathrm {cm}， $小圆盘的直径为$ d_2\ \mathrm {cm}.$
根据题意，得$ \pi(\frac {d_1}{2})^2-4 \pi(\frac {d_2}{2})^2=7 \pi$
即$ d_1^2-4\ \mathrm {d}_2^2=28.$
分解因式, 得$ (d_1+2\ \mathrm {d}_2)(d_1-2\ \mathrm {d}_2)=28$
$ ∵d_1+2\ \mathrm {d}_2 $均为整数,
$ ∴d_1+2\ \mathrm {d}_2， d_1-2\ \mathrm {d}_2 $均为整数.
又$ ∵d_1+2\ \mathrm {d}_2 $和$ d_1-2\ \mathrm {d}_2 $的奇偶性相同
$ ∴\{\begin{array}{l}d_1+2\ \mathrm {d}_2=14\\d_1-2\ \mathrm {d}_2=2\end{array}， $
解得$ \{\begin{array}{l}d_1=8\\d_2=3\end{array}.$
$ ∴\frac {d_1}{2}=4， \frac {d_2}{2}=1.5$
∴大、小圆盘的半径分别为$ 4\ \mathrm {cm} $和$ 1.5\ \mathrm {cm}.$
