In this thesis, the transient analysis of linearly coupled circuits using the finite element method is processed and the methods used for its analytical solution are described. In doing so, Heun’s time integration method and Theta method were used on a selected example of a circuit, and the created program was compared with the numerical simulation results from Simulink. The second chapter discusses in general the methods for harmonic analysis of linear current circuits in the time and frequency domain, of which the most practical method, the finite element method, stands out. The derivation for the branch of the linear coupled circuit for the Heun and Theta method is given in the third chapter and is used in the creation of the user program to define the input data and the mutual relations of the circuit elements. In the fourth chapter, a computer program created in MATLAB and used for linear circuit analysis is described. The aim of the program is to demonstrate the effectiveness of the finite element method in the modeling of circuit elements. In the last chapter, transient analysis is processed on a selected example of a circuit, the Heun and Theta methods are applied, and images of the solutions obtained by the computer program are shown for comparison with the numerical solution from Simulink.