  Usually such area of mathematics as differential equations acts as a consumer
of results given by functional analysis. This article will give an example of
the reverse interaction of these two fields of knowledge. Namely, the
derivation and study of integral transforms will be carried out using partial
differential equations. We will study generalised Weber-Orr transforms - its
invertibility theorems in $f\in L_1\cap L_2$, spectral decomposition,
Plancherel-Parseval identity. These transforms possess nontrivial kernel, so
spectral decomposition must involve not only continuous spectrum, but also
eigen functions which correspond to zero eigen value. We give a new approach to
the study of classical and generalized Weber-Orr transforms with a complete
derivation of the inversion formulas.
