Let $\bbK$ be an ordinary differential field with derivation $\partial$. Let $\cP$ be a system of $n$ linear differential polynomial parametric equations in $n-1$ differential parameters with implicit ideal $\id$. Given a nonzero linear differential polynomial $A$ in $\id$ we give necessary and sufficient conditions on $A$ for $\cP$ to be $n-1$ dimensional. We prove the existence of a linear perturbation $\cP_ϕ$ of $\cP$ so that the linear complete differential resultant $\dcres_ϕ$ associated to $\cP_ϕ$ is nonzero. A nonzero linear differential polynomial in $\id$ is obtained from the lowest degree term of $\dcres_ϕ$ and used to provide an implicitization algorithm for $\cP$.