a/b+b/a=-1; multiply through by ab: a^2+b^2=-ab so a^2+b^2+ab=0.
a^3-b^3=(a-b)(a^2+ab+b^2)=0 so a^3=b^3, a=b. But a/b=b/a=1, and 1+1 does not equal -1. Therefore there is no solution because a/b+b/a cannot equal -1.
(If a/b-b/a=-1, a^2-b^2+ab=0; b^2=a^2+ab; a^3-b^3=2a(a-b)(a+b)=2a(a^2-b^2).
If a/b+b/a+1, a^2+b^2-ab=0, ab=a^2+b^2 and a^3-b^3=2(a-b)(a^2+b^2).)