The paper considers the problem given by the boundary and additional condition for the quasi-linear
Mathieu differential equation with a parameter. The value of the desired function at the beginning point of the interval is
considered as an additional parameter and the desired function is replaced by the sum of the new unknown function and
the entered parameter. The Cauchy condition appears for quasi-linear ordinary differential equations. Employing the
solution to the Cauchy problem to the boundary condition, a system of quasi-linear algebraic equations in parameters is
formed. To find a solution for the system Newton’s iterative method is used. An algorithm for solving the problem under
consideration based on the D.S. Dzhumabaev parameterization method is proposed and various methods are applied to the numerical implementation of this algorithm. Steps of the algorithm include solving Cauchy problems. To solve
Cauchy problems the Bulirsh-Shter method, the Runge-Kutta method of the fourth order, and the Adams method are used
and these results are compared. Using various numerical or approximate methods, we get a new numerical or approximate
implementation of the algorithm. The accuracy of numerical solutions depends on the choice of methods for solving
Cauchy problems.