Solution
= Solution
Put $D_n=(-1)^ne^\pi-1$. The two <linear equations> from parts (i) and (ii) give $I_n=D_n+nJ_n$ and $J_n=-nI_n$, so $(1+n^2)I_n=D_n$. Therefore
$$
\boxed{I_n=\frac{(-1)^ne^\pi-1}{1+n^2},\qquad J_n=\frac{n\left[1-(-1)^ne^\pi\right]}{1+n^2}.}
$$
These formulas include $n=0$, when $I_0=e^\pi-1$ and $J_0=0$.