Let $\displaystyle f(t,x)=1-x.sin(t)$, for $\displaystyle (t,x) \in \mathbb{R}^2$.

(a)Show that $\displaystyle f$ satisfies a Lipschitz condition on $\displaystyle \mathbb{R}^2$ with respect to $\displaystyle x$. (Solved)

(b)Find the first 3 successive approximations $\displaystyle x_0(t)$, $\displaystyle x_1(t)$ and $\displaystyle x_2(t)$ of the initial value problem $\displaystyle x^{'}(t)=f(t,x)$, $\displaystyle x(0)=0$. (Solved)

(c)State the largest interval $\displaystyle I$ in which a unique solution to the initial value problem in (b) is defined. (Solved)

(d)Let $\displaystyle \phi(t)$ be the solution to the initial value problem in (b). Prove that $\displaystyle \vert \phi(t) \vert \leq e^2 \vert t \vert}$, for all $\displaystyle t \in I$.

I've difficulty in doing the last questions. How do I go about doing the last question?

Thanks in advance.