  A new method is introduced for studying boundary value problems for a class
of linear PDEs with {\it variable} coefficients. This method is based on ideas
recently introduced by the author for the study of boundary value problems for
PDEs with {\it constant} coefficients. As illustrative examples the following
boundary value problems are solved: (a) A Dirichlet and a Neumann problem on
the half line for the time-dependent Schr\"odinger equation with a space
dependent potential. (b) A Poincar\'e problem on the quarter plane for a
variable coefficient eneralisation of the Laplace equation.
