Let $y = f ( x )$ be the solution to the differential equation $\frac { d y } { d x } = x - y$ with initial condition $f ( 1 ) = 3$. What is the approximation for $f ( 2 )$ obtained by using Euler's method with two steps of equal length starting at $x = 1$ ?
(A) $- \frac { 5 } { 4 }$
(B) 1
(C) $\frac { 7 } { 4 }$
(D) 2
(E) $\frac { 21 } { 4 }$

We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. For all $r \geqslant 0$, we denote $B_r$ the closed ball with center $y_{\text{init}}$ and radius $r$.
Show that we can choose $r > 0$ and $T > 0$ such that $B_r \subset \Omega$ and such that for all $N \in \mathbb{N}^*$, we can define by recursion, by setting $\Delta t = \frac{T}{N}$, a sequence $(y_n)_{0 \leqslant n \leqslant N}$ taking values in $B_r$ such that: $$y_0 = y_{\text{init}}, \quad y_{n+1} = y_n + \Delta t F(y_n), \forall n \in \{0, \cdots, N-1\}$$