There is actually a unified treatment of transform theory. This is what chapter seven of
Keener's book is about.

There is a relationship between Linear differential equations, spectrum,
Green functions, contour integration, Dirac Deltas, and Transforms.

The story goes like this. Given a linear operator (differential) $L$,
form the operator $L - \lambda = 0$, with given boundary conditions,
find its Green function $G(x, \xi, \lambda)$. Then

\begin{equation}
\delta(x - \xi) = -\frac{1}{2 \pi \mathrm{i}} \int_{C_{\infty}}
G(x, \xi , \lambda) \, d \lambda,
\end{equation}

where the contour $C_{\infty}$ is a circle having all the spectrum of $L$ inside.

This particular Dirac delta representation is the product of the direct and inverse transforms.

Here are a few examples:

**Sine transform pair:**
Functions continuously differentiable in $[0,1]$.
\begin{equation}
Lu = - u'' \quad , \quad u(0)=u(1)=0.
\end{equation}

result:

\begin{eqnarray*}
U_k &=& 2 \int_0^1 d \xi \sin( k \pi \xi) u( \xi ) \\
u(x) &=& \sum_{k=1}^{\infty} U_k \sin k \pi x.
\end{eqnarray*}

**Cosine transform pair:**
\begin{equation}
Lu = - u'' \quad , \quad u'(0)=u'(1)=0.
\end{equation}

result:

\begin{eqnarray*}
U_k &=& 2 \int_0^1 d \xi \cos( k \pi \xi) u( \xi ) \\
u(x) &=& \sum_{k=1}^{\infty} U_k \cos k \pi x.
\end{eqnarray*}

**Sine transform integral**

Functions in $L^2[0, \infty)$.
\begin{equation}
Lu = - u'' \quad , \quad u(0)=0 \quad , \quad \lim_{x \to \infty} u(x) = 0.
\end{equation}

result:

\begin{eqnarray*}
U(\mu) &=& \frac{2}{\pi} \int_0^{\infty} dx u(x) \sin \mu x \\
u(\xi) &=& \ \int_0^{\infty} d \mu U(\mu) \sin \mu x \\
\end{eqnarray*}

**cosine transform integral**
\begin{equation}
Lu = - u'' \quad , \quad u'(0)=0 \quad , \quad \lim_{x \to \infty} u(x) = 0.
\end{equation}

result:

\begin{eqnarray*}
U(\mu) &=& \frac{2}{\pi} \int_0^{\infty} dx u(x) \cos \mu x \\
u(\xi) &=& \ \int_0^{\infty} d \mu U(\mu) \cos \mu x \\
\end{eqnarray*}

**Fourier transform**
Functions in $L^2(-\infty, \infty)$.
\begin{equation}
Lu = - u'' \quad , \quad , \quad \lim_{x \to \pm \infty} u(x) = 0.
\end{equation}

result:

\begin{eqnarray*}
U(\mu) &=& \int_{-\infty}^{\infty} dx \; u(x) \; \mathrm{e}^{\mathrm{i} \mu \xi} \\
u(\xi) &=& \frac{1}{2 \pi} \int_{-\infty}^{\infty} U(\mu) \mathrm{e}^{-\mathrm{i} \mu \xi}
d \mu.
\end{eqnarray*}

The list follows: Mellin, Hankel, etc. Keener shows the linear operators with boundary conditions for these.

I do not know which operator and boundary conditions generate the Laplace
transform. I opened this question in StackExchange Mathematics for this particular problem.

I am writing some notes about this matter
here.

**UPDATE:**
I solved the problem of connecting the Laplace transform to an ODE with boundary conditions. Please see here

Thanks.