The coordinate transformation and the exact solutions of the Schr\"{o}dinger equation with position-dependent effective mass

Using the coordinate transformation method, we solve the one-dimensional Schr\"{o}dinger equation with position-dependent mass(PDM). The explicit expressions for the potentials, energy eigenvalues and eigenfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hemite and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term $V_m$ which produced from the dependence of mass on the coordinate, compared with that for the systems of constant mass. The properties of $V_m$ for several mass functions are discussed.