1(c)

In one step of the Improved Euler method we have m₁ = f(x_j, y_j) = 3 y_j, m₂ = f(x_j+h, y_j + h m₁) = 3 (1 + 3 h) y_j, and y_j+1 = y_j + h (m₁ + m₂)/2 = (1 + 3 h + 9 h²/2) y_j. On the other hand, $\psi_j(x_{j+1}) = {\rm e}^{3 h} y_j$. From the Taylor series, ${\rm e}^{3 h} = 1 + 3 h + 9 h^2/2 + O(h^3)$, so the local error is $e_{j+1} = \psi_j(x_{j+1}) - y_{j+1} = y_j O(h^3) = O(h^3)$.