$\begin{aligned} & f(1)=1, f(a)=0 \\ & f^2(x)=\operatorname{Lim}_{r \rightarrow x}\left(\frac{2 r^2\left(f^2(r)-f(x) f(r)\right)}{r^2-x^2}-r^3 e^{\frac{f(r)}{r}}\right) \\ & =\operatorname{Lim}_{r \rightarrow x}\left(\frac{2 r^2 f(r)}{r+x} \frac{(f(r)-f(x))}{r-x}-r^3 e^{\frac{f(r)}{r}}\right) \\ & f^2(x)=\frac{2 x^2 f(x)}{2 x} f^{\prime}(x)-x^3 e^{\frac{f(x)}{x}} \\ & y^2=x y \frac{d y}{d x}-x^3 e^{\frac{y}{x}} \\ & \frac{y}{x}=\frac{d y}{d x}-\frac{x^2}{y} e^{\frac{y}{x}} \\ & \text { Put } y=v x \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x} \\ & v=v+x \frac{d v}{d x}-\frac{x}{v} e^v \\ & \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{\mathrm{e}^{\mathrm{v}}}{\mathrm{v}} \Rightarrow \mathrm{e}^{-\mathrm{v}} \mathrm{vdv}=\mathrm{dx} \\ & \operatorname{Integrating} \text { both side } \\ & \mathrm{e}^{\mathrm{v}}(\mathrm{x}+\mathrm{c})+1+\mathrm{v}=0 \\ & f(1)=1 \Rightarrow x=1, y=1 \Rightarrow \mathrm{c}=-1-\frac{2}{\mathrm{e}} \\ & \mathrm{e}^{\mathrm{v}}\left(-1-\frac{2}{\mathrm{e}}+\mathrm{x}\right)+1+\mathrm{v}=0 \\ & e^{\frac{y}{x}}\left(-1-\frac{2}{e}+x\right)+1+\frac{y}{x}=0 \\ & x=a, y=0 \Rightarrow a=\frac{2}{e} \\ & \mathrm{ae}=2\end{aligned}$