Two conditions on the second order terms fix b and c

Let $f : [0, 1] \times \mathbb{R} \to \mathbb{R}$ be a function such that f and its partial derivatives of orders less than or equal to 3 are continuous and bounded. Let $y : [0, 1] \to \mathbb{R}$ be the solution of dy/dt $= f(t, y(t)), 0 \le t \le 1, y(0) = y_{0}$, where $y_{0} \in \mathbb{R}$. For h > 0, denote t_j = jh. Let Y_j be an approximation of y(t_j), defined by Y_(j+1) = Y_j + ahf(t_j, Y_j) + bhf(t_(j+1), Y_j + chf$(t_j, Y_j)), 0 < (j+1)h < 1, Y_{0} = y_{0}$, where $a, b, c \in \mathbb{R}$. If there exists an M > 0 such that |y(t_(j+1)) − y(t_j) − ahf(t_j, y(t_j)) − bhf(t_(j+1), y(t_j) + chf(t_j, y(t_j)))| ≤ Mh$^{3}$ for every h > 0, and every j with 0 ≤ (j+1)h < 1, then