AHSME 1967 Problem 10

If $\frac{a}{10^x-1}+\frac{b}{10^x+2}=\frac{2 \cdot 10^x+3}{(10^x-1)(10^x+2)}$ is an identity for positive rational values of $x$, then the value of $a-b$ is: $\textbf{(A)}\ \frac{4}{3} \textbf{(B)}\ \frac{5}{3} \textbf{(C)}\ 2 \textbf{(D)}\ \frac{11}{4} \textbf{(E)}\ 3$