A Question about Non Linear Differential Equation

[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.

 

[blue_raver22]

I understand your question about finding the exact solution for a non linear differential equation. The process of finding the exact solution for a non linear differential equation can be challenging and there is no one specific method that can be applied to all cases. However, there are some common approaches that can be used to solve non linear differential equations.

One approach is to use numerical methods such as Euler's method or Runge-Kutta method to approximate the solution. These methods use step-by-step calculations to approximate the solution, but they may not always give an exact solution.

Another approach is to use analytical methods, such as separation of variables, substitution, or power series expansion. These methods can be applied to specific types of non linear differential equations and may lead to an exact solution.

Additionally, there are computer software programs available that can solve non linear differential equations numerically or symbolically, such as Mathematica or Maple.

I would recommend consulting a textbook or online resources for specific methods and techniques for solving non linear differential equations. It is also helpful to seek guidance from a mathematics or physics professor or a colleague who has experience in solving non linear differential equations.

In conclusion, finding the exact solution for a non linear differential equation can be a complex process and may require various methods and techniques. I suggest exploring different approaches and seeking guidance from experts in the field to find the best solution for your specific equation.