 Solution

Insert the exact solution and expand about $t=t_{n+2}$. Since $f(y)=y^{\prime }$ and $f^{\prime }(y)f(y)=y^{\prime \prime }$, the left side isThe coefficient of $h^j{y}^{(j)}$ on the right side isfor $j\ge 1$, with the last term absent for $j=0$. These coefficients agree through $j=3$; at $j=4$ the left side minus the right side is $1/10$. Thus the local truncation error is $h^4{y}^{(4)}/10+O(h^5)$ and the method has order three.