The equation $dx/dt=-3x+2$, $x(0)=1$, is solved by forward Euler. The largest time step that can be used without making the numerical solution unstable is ________.
Forward Euler gives $x_{n+1}=(1-3h)x_n+2h$. Subtract the equilibrium 2/3 to obtain the error recursion $e_{n+1}=(1-3h)e_n$.
For bounded errors, $|1-3h|\le1$, giving $0\le h\le2/3$. Thus the limiting step is **2/3≈0.6667**. At the endpoint the error alternates without decaying; strict asymptotic decay requires h<2/3. The official range includes this boundary value.
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