General Exact Solutions Of The Special Second-order Singular Algebraic Differential Equations

Second-order homogeneous algebraic differential equations arise in a widevariety of scientific and engineering applications,including physical and technicalproblems,computer-aised design.And we can investigate the problem by solving thesolutions of the singular algebraic differential equation or discussing the propertiesof the solutions.Therefore,it is necessary to study the second-order homogeneousalgebraic differential equations.The second-order homogeneous algebraic differential equation can be statedas,where are known such that is nonsingular for some.In this papar,we derive aformula for the general exact solutions of the above equation under the conditionthat there exists with,such that.For a special second-order homogeneous algebraicdifferential equation coming form a physics model investigated by Bhat andBernstein,we give a detailed description about how to construct a particular solutionY with ind(Y)=1of the latter quadratic matrix equation.Some numerical examples areprovided as well.