# What are all types of elementary second order ordinary differential equation that can not be expressed in closed form?

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!

• By 'define' presumably you want something like an algorithm; in other words, an analog to the Risch algorithm that covers arbitrary second-order differential equations rather than just the elementary antiderivative? – Steven Stadnicki May 1 '15 at 21:19
• @StevenStadnicki No, this question ask how to distinguish which types could solve in closed form and which is not, for example, homogeneous linear ODE is one type that we could solve – Victor May 1 '15 at 21:26
• Ahhh, so something closer to Liouville's Theorem then? It sounds like what you want is essential differential Galois theory... – Steven Stadnicki May 1 '15 at 21:33
• @StevenStadnicki: Almost. It's precisely what Sophus Lie had in mind when he invented his Lie Groups: sort of Galois theory for differential equations. That has been the original setting, if I'm informed well. – Han de Bruijn May 3 '15 at 14:12