smc

We can factor $a^3-b^3=19x^3$ into: $$(a-b)(a^2+ab+b^2)=19x^3.$$ Substituting yields: $$x(a^2+ab+b^2)=19x^3$$ $$a^2+ab+b^2=19x^2.$$ This is equal to: $$(a-b{)}^2+3ab=19x^2$$ $$x^2+3ab=19x^2$$ $$ab=6x^2.$$ Checking with the possible answers, along with $a-b=x$ yields the only answer to be $\boxed{a=3x\text{ or }a=-2x}.$