Chapter 13.4, Problem 1E

In Problems 1-6, let $\varphi \left(x,y_0\right)$ be the solution to the given differential equation that satisfies $y\left(0\right)=y_0$ and let $\varphi \left(x,{\tilde{y}}_0\right)$ be the solution satisfying $y\left(0\right)={\tilde{y}}_0$. Assume the solutions exist on $\left[-1,1\right]$ and their graphs remain in the rectangle $R_0=\left\{\left(x,y\right):-1\le x\le 1,-4\le y\le 4\right\}$. Use Theorem 9 to find a bound on the difference between $\varphi \left(x,y_0\right)$ and $\varphi \left(x,{\tilde{y}}_0\right)$ for $\text{x}$ in $\left[-1,1\right]$ if $\left|y_0-{\tilde{y}}_0\right|<{10}^{-2}$.

$y^{\prime }=\sin \left(x+y\right)$