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