Can we define all types of elementary second order ordinary differential equation that can not be expressed in closed form as opposed to the one that we can solve?
In differential algebra, Picard–Vessiot theory is the study of the differential field extension generated by the solutions of a linear differential equation, what about other types, say elementary second order nonlinear ordinary differential equation?
Elementary second order ordinary differential equations consist of any power of the typical x and y functions, exponential equations, logarithmic equations and trigonometric functions, inverse hyperbolic functions and the complex number i.
closed form is defined as "It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), and complex number i, but usually no limit." also it is an finitary expression
edit 1 - Since the bounty will end in 3 days, someone please answer it quick!