The Basic Research On Solving The Second-order Linear Ordinary Differential Equation With Variable Coefficients And Maple's Applications On It

Linear ordinary differential equation of the second order d~2y/dx~2+p(x)dy/dx+Q(x)y=0has a wide application in the science and technology. Especially in Physics, linear ordinarydifferential equation of the second order and its eigenvalue problem are important base ofsolving Mathematical Physics equation. A variety of Physical problems belong to solvinglinear ordinary differential equation of the second order. Solving the second-order linearordinary differential equation with variable coefficients, however, is quite difficult. Nowadaysthere has not been a comparatively universal way. Usually adopting the series method onlycan get local solution in the neighborhood of some point, also solution of infinite series orapproximate solution, which is not convenient for analysis in theory. Therefore, manyPhysical teaching staff take interests in solving the second-order linear ordinary differentialequation with variable coefficients.In the theory of linear ordinary differential equation of the second order, equations withconstant coefficients are solvable and the property of special function equation has been madea deep research, therefore, which can be regarded as "solvable equation". However, whensolving the practical problems, we usually meet strange equation with variable coefficients,which is comparatively difficult to solve.On the research of differential equation and the theory of special function, this thesisprovides a new way of changing some second-order linear ordinary differential equation withvariable coefficients into known special solvable equation, and converted relations betweenthese types of equations. Meanwhile in substance, it expatiates the way of changing thevariable coefficients of the equation into constants, and sets forth the basic thought andprocedures of solving the second-order linear ordinary differential equation with variablecoefficients. Finally, with Maplesoft, we can compile the complicated solving procedures intoMaple program, and simplify the complicated calculation in the transformed course ofequation with the ability of mighty symbolic calculation, which may make the solution of thesecond-order linear ordinary differential equation with variable coefficients simplified, and bein favor of both studying and researching the differential equation.