<p>A linear dust acoustic wave has been studied in an inhomogeneous dusty plasma, taking into account the presence of nonthermal ions. The hydrodynamic model is considered to study the wave dynamics. Due to density inhomogeneities, a variable-coefficient second-order ordinary differential equation is derived. Theoretical and numerical results discover the formation of amplifying linear waves due to gradient instability. Due to the presence of Cairns-distributed nonthermal ions, a modification to the theoretical result obtained using the WKB approximation has been observed for the linear dust acoustic wave in a multisized inhomogeneous dusty plasma. The effect of nonthermal ions on wave propagation in the region has been investigated. The numerical solution of the variable coefficient evolution equation is in good agreement with the analytical results obtained by the WKB approximation, especially in the presence of a smaller number of nonthermal (fast) ions in a slowly varying medium. The effect of the ions-to-electrons temperature ratio on the wavelength of the linear dust acoustic wave is also observed.</p>