Chapter 13.4, Problem 14E

To determine

To show:

The following result remains valid if $L$ is replaced by a positive continuous function $l\left(x\right)$ and the inequality $\left|\varphi \left(x,y_0\right)-\varphi \left(x,{\tilde{y}}_0\right)\right|\le \left|y_0-{\tilde{y}}_0\right|e^{Lh}$ is replaced by the inequality $\left|\varphi \left(x,y_0\right)-\varphi \left(x,{\tilde{y}}_0\right)\right|\le \left|y_0-{\tilde{y}}_0\right|\exp \left\{\left|{\displaystyle \underset{x_0}{\overset{x}{\int }}l\left(x\right)dx}\right|\right\}$.

Let $f$ and $\partial f/\partial y$ be continuous functions on an open rectangle

$R=\left\{\left(x,y\right):a<x<b,c<y<d\right\}$

containing the point $\left(x_0,y_0\right)$. Assume that for all ${\tilde{y}}_0$ sufficiently close to $y_0$, the solution $\varphi \left(x,{\tilde{y}}_0\right)$ to $(7)$ exists on the interval $\left[x_0-h,x_0+h\right]$ and its graph lies within a fixed closed rectangle $R_0\subset R$. Then, for $\left|x-x_0\right|\le h$,

$\left|\varphi \left(x,y_0\right)-\varphi \left(x,{\tilde{y}}_0\right)\right|\le \left|y_0-{\tilde{y}}_0\right|e^{Lh}$

Here, $L$ is any positive constant such that $\left|\left(\partial f/\partial y\right)\left(x,y\right)\right|\le L$ for all $\left(x,y\right)$ in $R_0$.