Abstract The scaled boundary finite-element equations in displacement and dynamic stiffness, which are ordinary differential equations, derived in the accompanying paper involve the discretization of the boundary only. The general solution procedure is demonstrated addressing an illustrative example which consists of a two-dimensional out-of-plane (anti-plane) motion with a single degree of freedom on the boundary. For statics and dynamics in the frequency domain, the displacements in the domain and the stiffness matrix with degrees of freedom on the boundary only are obtained analytically for bounded and unbounded media. The radiation condition is satisfied exactly using the high-frequency asymptotic expansion for the dynamic-stiffness matrix of an unbounded medium. The mass matrix for a bounded medium is determined analytically. Body loads in statics are calculated analytically. Numerical procedures to calculate the dynamic-stiffness and unit-impulse response matrices for an unbounded medium are also presented. The scaled boundary finite-element method is semi-analytical as the ordinary differential equations in displacement are solved analytically, which permits an efficient calculation of displacements, stresses and stress intensity factors. This boundary-element method based on finite elements leads to a reduction of the spatial dimension by one. As no fundamental solution is required, no singular integrals are evaluated and anisotropic material is analysed without additional computational effort.