5. Given the correspondence of points in a Cartesian coordinate system

rule $f: P(m, n) \rightarrow P^{\prime}(\sqrt{m}, \sqrt{n})(m \geqslant 0, n \geqslant$ 0 ). If a segment of a curve corresponds to an arc of an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(x \geqslant 0, y \geqslant 0)$ under the correspondence rule $f$, then the equation of this segment of the curve is
5. $y=b^{2}\left(1-\frac{x}{a^{2}}\right)\left(0 \leqslant x \leqslant a^{2}\right)$.

Let the curve equation be $y=f(x)(s \leqslant x \leqslant t)$, then the point $P(x, f(x))$ on the curve corresponds to the point $P^{\prime}(\sqrt{x}, \sqrt{f(x)})$ on an arc of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(x \geqslant 0, y \geqslant 0)$. Therefore,
$$
\frac{x}{a^{2}}+\frac{f(x)}{b^{2}}=1(x \geqslant 0, f(x) \geqslant 0),
$$

which means $f(x)=b^{2}\left(1-\frac{x}{a^{2}}\right)\left(0 \leqslant x \leqslant a^{2}\right)$.