Chapter 13.RP, Problem 12RP

Let $\varphi \left(x\right)$ be the solution to $y^{\prime }=x\sin y$, $y\left(0\right)=y_0$, and let $\phi \left(x\right)$ be the solution to $y^{\prime }=xy$, $y\left(0\right)=y_0$. Assume the solutions exist on $\left[-1,1\right]$ and their graphs remain in the rectangle $R_0=\left\{\left(x,y\right):-1\le x\le 1,-\pi \le y\le \pi \right\}$. Find a bound on the difference between $\varphi \left(x\right)$ and $\phi \left(x\right)$ for $\text{x}$ in $\left[-1,1\right]$.