Let $X$ be a regular one-dimensional transient diffusion and $L^y$ be its local time at $y$. The stochastic differential equation (SDE) whose solution corresponds to the process $X$ conditioned on $[L^y_{\infty}=a]$ for a given $a\geq 0$ is constructed and a new path decomposition result for transient diffusions is given. In the course of the construction of the SDE the concept of {\em recurrent transformation} is introduced and {\em Bessel-type motions} as well as their SDE representations are studied. A remarkable link between an $h$-transform with a minimal excessive function and recurrent transformations is found, which, as a by-product, gives a useful representation of last passage times as a mixture of first hitting times. Moreover, the Engelbert-Schmidt theory for the weak solutions of one dimensional SDEs is extended to the case when the initial condition is an entrance boundary for the diffusion. This extension was necessary for the construction of the Bessel-type motion which played an essential part in the SDE representation of $X$ conditioned on $[L^y_{\infty}=a]$.