Existence and uniqueness analysis

\[\begin {aligned} x^{\prime \prime }-x&={\mathrm e}^{t}\\ x \left (0\right ) &= 0\\ x^{\prime }\left (0\right ) &= 1\\ \end {aligned}\]

This is a linear ODE. In canonical form it is written as

\begin{align*} x^{\prime \prime } + p(t)x^{\prime } + q(t) x &= F \end{align*}

Comparing the above to the given ODE shows that

\begin{align*} p(t) &=0\\ q(t) &=-1\\ F &={\mathrm e}^{t} \end{align*}

Hence the ode is

\begin{align*} x^{\prime \prime }-x = {\mathrm e}^{t} \end{align*}

The domain of \(p(t)=0\) is

\begin{align*} \{-\infty <t <\infty \} \end{align*}

And the point \(t_0 = 0\) is inside this domain. The domain of \(q(t)=-1\) is

\begin{align*} \{-\infty <t <\infty \} \end{align*}

And the point \(t_0 = 0\) is also inside this domain. The domain of \(F ={\mathrm e}^{t}\) is

\begin{align*} \{-\infty <t <\infty \} \end{align*}

And the point \(t_0 = 0\) is also inside this domain. Therefore solution exists and is unique.