Existence and uniqueness analysis

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

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 &=\cos \left (t \right ) \end{align*}

Hence the ode is

\begin{align*} x^{\prime \prime }+x = \cos \left (t \right ) \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 \(F =\cos \left (t \right )\) 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.