Flexural deflections of several plates and beams under an unknown transverse, concentrated, time-dependent force are solved for various edge conditions. The consideration of displacements and the use of Hertz’s law of impact at the point of contact lead to a nonlinear integral equation for the contact force in all cases of transverse impact. Two methods are introduced to treat this equation: (a) Generalized Galerkin method; (b) collocation method. Method (a) is the generalization of the well-known Galerkin method which is suitable for the problems in which part of the boundary is unknown in advance, while certain conditions are given there. The method is applicable to a very large class of differential and integral equations. The collocation method leads to a quick, reasonable, approximate solution. Various auxiliary curves in both cases reduce the solution to a routine. Examples are worked out and plotted for various beams and plates. Deflections are plotted.