Calculate difference equation from impulse response
We know that the unit impulse response is the transfer function, hence we can write:

\$\dfrac{Y(z)}{X(z)}= H(z) = \dfrac{z^{-1} + \frac{1}{2} z^{-2}}{1-\frac{3}{5} z^{-1} + \frac{2}{25} z^{-2}}\$

where \$X(z)\$ and \$Y(z)\$ are the input and output signals, respectively.

Cross-multiplying:

\$Y(z) (1-\frac{3}{5} z^{-1}+\frac{2}{25} z^{-2}) = X(z) (z^{-1}+\frac{1}{2} z^{-2})\$

Inverse z-transforming and re-arranging:

\$y[n]= x[n-1]+\frac{1}{2} x[n-2] + \frac{3}{5} y[n-1]-\frac{2}{25}y[n-2]\$