The solution of ordinary differential equations with the help of Chebyshev polynomials

In this study, the numerical solutions of nonhomogenous linear ordinary differential equations with constant coefficient or variable coefficient without initial and boundary conditions are given with Chebyshev approximation methods in terms of Chebyshev polynomials of the first, second, third and fourth kinds. Chebyshev series expansion y(x)= Sum(i=0 to n)q_i(x)*(fi)_i(x) ,(fi)_i(x) (i>=0) is equal to Chebyshev polynomials of the first kind or second kind or third kind or fourth kind, is used in these methods. The solutions of nonhomogenous linear ordinary differential equations with constant coefficient and variable coefficient are obtained from derivatives of Chebyshev polynomials and representation in terms of Chebyshev polynomials of x power n (n>=0). Furthermore, the equations are given to finding coefficients of approximation polynomial of nonhomogenous the first and second order linear ordinary differential equations with constant coefficients.