Finding exact solutions from known symmetries of nonlinear differential equations

[yicong2011]

Given a non linear differential equation, we have know the symmetry of its solution. How can we finally get the exact solution? Any methods or reference?

(I am a physics students; recently I met with something relevant to non linear differential equation.)

 

[HallsofIvy]

How, or if, one can solve a non-linear differential equation strongly depends upon the exact form of the equation. "Almost all" non-linear equations cannot be solved in terms of elementary functions.

 

[Eynstone]

Given a non linear differential equation, we have know the symmetry of its solution. How can we finally get the exact solution? Any methods or reference?QUOTE]
The theory of Lie groups can be employed to solve an equation using the symmetry.

 

[Studiot]

You might sketch/plot the direction fields of the equation.