Example 2. If $a, b$ are both positive real numbers, and $\frac{1}{a}-\frac{1}{b}-$ $\frac{1}{a+b}:=0$, then $\left(\frac{b}{a}\right)^{3}+\left(\frac{a}{b}\right)^{3}=$ $\qquad$
Given that both sides are multiplied by $a+b$, we know $\frac{b}{a}-\frac{a}{b}=1$. Considering its dual form $\frac{b}{a}+\frac{a}{b}=A$, adding the two equations gives $\frac{2 b}{a}$ $=A+1$, and subtracting them gives $\frac{2 a}{b}=A-1$. Multiplying the last two equations yields $A^{2}$ $-1=4, A=\sqrt{5}(\because A>0)$, the original equation $=A^{3}-3 A$ $=2 \sqrt{5}$.