Question

In Problems 9 and 10 find an interval centered about $\mathbf{x}\mathbf{=}\mathbf{0}$for which the given initial-value problem has a unique solution.

$\mathbf{y}\mathbf{\text{'}\text{'}}\mathbf{+}\left(\tan x\right)\mathbf{y}\mathbf{=}{\mathbf{e}}^{\mathbf{x}}\mathbf{,}\mathbf{y}\left(0\right)\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{0}$

Short answer

The interval for the unique solution is$\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$

Step-by-step solution

Define general second-order linear differential equation and the Theorem for Existence of a Unique Solution

A general second-order linear differential equation is defined as ${\mathrm{a}}_2\left(\mathrm{x}\right)\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}+{\mathrm{a}}_1\left(\mathrm{x}\right)\frac{\mathrm{dy}}{\mathrm{dx}}+{\mathrm{a}}_0\left(\mathrm{x}\right)\mathrm{y}=\mathrm{g}\left(\mathrm{x}\right)$.

Consider${\mathrm{a}}_{\mathrm{n}}\left(\mathrm{x}\right),{\mathrm{a}}_{\mathrm{n}-1}\left(\mathrm{x}\right),...,{\mathrm{a}}_1\left(\mathrm{x}\right),{\mathrm{a}}_0\left(\mathrm{x}\right)$ and$\mathrm{g}\left(\mathrm{x}\right)$ be continuous on an interval I and let${\mathrm{a}}_{\mathrm{n}}\left(\mathrm{x}\right)\ne 0$ for every value of x in this interval. If there is any point$\mathrm{x}={\mathrm{x}}_0$ in this interval, then there exists a solution$\mathrm{y}\left(\mathrm{x}\right)$ of this initial-value problem on the interval and is unique. This is known as the Theorem for Existence of a Unique Solution.

Find the interval in which the functions are continuous

Comparing the given differential equation with the general second-order linear differential equation,

$\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}+\left(\tan \mathrm{x}\right)\mathrm{y}={\mathrm{e}}^{\mathrm{x}}$

The functions are:

$$\begin{gathered} {\mathrm{a}}_2\left(\mathrm{x}\right)=1 \\ {\mathrm{a}}_1\left(\mathrm{x}\right)=0 \\ {\mathrm{a}}_0\left(\mathrm{x}\right)=\tan \mathrm{x} \\ \mathrm{g}\left(\mathrm{x}\right)={\mathrm{e}}^{\mathrm{x}} \end{gathered}$$

The functions ${\mathrm{a}}_2\left(\mathrm{x}\right),{\mathrm{a}}_1\left(\mathrm{x}\right),\mathrm{g}\left(\mathrm{x}\right)$continuous in the interval $\left(-\infty ,\infty \right)$and the function ${\mathrm{a}}_0\left(\mathrm{x}\right)$is continuous in the intervals$...,\left(-\frac{5\mathrm{\pi }}{2},-\frac{3\mathrm{\pi }}{2}\right),\left(-\frac{3\mathrm{\pi }}{2},-\frac{\mathrm{\pi }}{2}\right),\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right),\left(\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{2}\right),.....$.

Out of these intervals, the interval centered about$x=0$ is $\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$.

Find the interval for unique solution

All the functions are continuous in the interval for ${\mathrm{a}}_2\left(\mathrm{x}\right)\ne 0$, true for all values of $\mathrm{x}$.

Using the Theorem for Existence of a Unique Solution,

$\mathbf{Hence}\mathbf{,}\mathbf{}\mathbf{the}\mathbf{}\mathbf{interval}\mathbf{}\mathbf{for}\mathbf{}\mathbf{the}\mathbf{}\mathbf{initial}\mathbf{-}\mathbf{value}\mathbf{}\mathbf{problem}\mathbf{}\mathbf{having}\mathbf{}\mathbf{a}\mathbf{}\mathbf{unique}\mathbf{}\mathbf{solution}\mathbf{}\mathbf{is}\mathbf{}\mathbf{(}\mathbf{-}\frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{,}\frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{)}$