Solution
= Solution
The <characteristic equation> is $r^2+r-2=(r-1)(r+2)=0$, giving $y=Ae^x+Be^{-2x}$. The <initial conditions> give $A+B=1$ and $A-2B=-1$. Solving yields $A=1/3$, $B=2/3$, hence
$$
\boxed{y(x)=\frac13e^x+\frac23e^{-2x}}.
$$
The two distinct characteristic roots supply all solutions of this second-order <linear ordinary differential equation>.