Ordinary Differential Equations

The techniques for solving ordinary differential equations (ODE) are of fundamental importance in the theory of differential equations. The solution of ODEs is of fundamental importance in solving partial differential equations (PDE). For first order ODEs, this chapter considers separable equations, homogeneous type equations, exact ODEs, integrable condition, integrating factors, the Stokes method for homogeneous type, the Jacobi method, the Euler method, standard linear form, the Bernoulli equation, the Riccati equation, integration by differentiation, the Clairaut equation, singular solution, the Lagrange equation, factorization of nonlinear form, and Taylor series expansion. For second order ODE with non-constant coefficients, the chapter considers the Euler equation, Laplace type equation, Liouville problem, Mainardi approach for Liouville problem, and finally the Liouville transformation. Both homogeneous and nonhomogeneous forms are considered. For higher order ODEs, the chapter considers the Euler equation of order n, adjoint differential equation of an n-th order ODE, Sarrus method, rule of transformation, and homogeneous equation.