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It seems that we cannot solve all differential equations that bump up into nature, and here I mean finding a general or analytical solution, not just numerical approximation (P.S: I'm only referring to the cases where a solution was proven to exist, I'm aware that some diff equation have no solution, so that's not the case of my discussion here).

1. Give me a list of ODE that we still don't know how to solve and a method is waiting to be invented/discovered, or maybe we need new mathematics to emerge in order to solve these ODEs.

These are the type of Ordinary Differential Equations and their corresponding methods of solution:

  • Separable equations (solution by integration)
  • Linear equations (integrating factor method)
  • Exact equations (differential of a function of two variables)
  • Other (Solution by substitution)

More methods: SERIES SOLUTIONS OF LINEAR EQUATIONS and THE LAPLACE TRANSFORM

It seems that all these types of ODE have its own methods to solve them, so the scenario that I could imagine is that some nonlinear equations still cannot be solved by any method.

2. Can you extend the list above by citing more methods used in solving ODEs?

3. Are mathematicians trying to find a general method to solve nonlinear ODEs?

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There are many more methods for finding closed-form solutions to differential equations.
Symmetry-based methods are quite powerful, especially when implemented in computer algebra systems, and are still an active area of research. For example, you might look at Bluman and Kumei, "Symmetries and Differential Equations".

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