Vallipalayam Chandrasekara, Muthusamy Lakshmananb, Karlygash Dosmagulovac,d,e, Zhanat Zhunussovac
aCentre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed University, Thanjavur – 613 401, India
bCentre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
cDepartment of Mathematics, Al-Farabi Kazakh National University, Almaty, Kazakhstan
dInstitute of Mathematics and Mathematical Modeling MES RK, Almaty, Kazakhstan
eDepartment of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
In this work, we develop a systematic procedure to obtain the general solutions of a class of nonlinear ordinary differential equations (ODEs) directly through special functions or other known functions. By introducing a suitable transformation in the state variable/dependent variable of the given nonlinear ODE, we can relate it to one of the special function equations, including Hermite’s equation, Legendre’s equation, and Laguerre’s equation, or other equations solvable through known functions, including isochronous and limit cycle solutions. This procedure can be further generalized to higher-order nonlinear ODEs. Obtaining the general solutions of the nonlinear ODEs with the help of special functions is new to the literature to our knowledge.
Keywords: Nonlinear ordinary differential equations; Special functions; Hermite equation; Legendre’s equation; Laguerre’s equation.
Chandrasekar, V., Lakshmanan, M., Dosmagulova, K., & Zhunussova, Z. (2023). Direct method of solving nonlinear ordinary differential equations through known functions. Advances in the Theory of Nonlinear Analysis and Its Application, 7(5), 130–140.