Solvability v.s. the existance of a solution for nonlinear ODE's

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In summary, there is no analytical way to solve most nonlinear differential equations and as a result, computers are often used to solve them numerically. While it is assumed that no analytical solution exists for these equations, it is possible that a well-defined function could satisfy them, though it is unlikely due to the limited subset of functions that have closed form representations.
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pergradus
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In physics we're told often that there is no analytical way to solve most nonlinear differential equations (I don't know if this can be proven or not, or it's just assumed because no one has found a way to do it), so we use a computer to solve them numerically.

I'm wondering though, assuming it is impossible to solve a differential equation by analytical means, does that mean that there is no analytical solution which exists? That is, could you somehow guess some well-defined function that satisfies the nonlinear equation, even if there's no way of directly solving the equation for it?
 
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In many cases (most, actually) there just doesn't exist a solution in closed form. The functions you're used to (i.e. functions that have closed form representations) are only a very small subset of all the possible functions; it's natural that they would satisfy only a small subset of PDE's and ODE's.
 

1. What is the difference between solvability and the existence of a solution for nonlinear ODE's?

Solvability refers to the ability to find an explicit formula or algorithm to solve a nonlinear ODE, while the existence of a solution simply means that a solution exists for the given ODE.

2. What types of nonlinear ODE's are generally considered to be solvable?

Nonlinear ODE's that can be reduced to a separable form, exact form, or first-order linear form are typically considered to be solvable.

3. Can all nonlinear ODE's be solved using numerical methods?

Yes, numerical methods such as Euler's method or Runge-Kutta methods can be used to approximate solutions for all nonlinear ODE's. However, these methods may not provide an exact solution and may require significant computational resources.

4. How does the nonlinearity of an ODE affect its solvability?

In general, the more nonlinear an ODE is, the more difficult it is to find an explicit solution. This is because nonlinear equations do not have a linear superposition property, making them more complex to solve compared to linear equations.

5. Are there any techniques that can be used to solve particularly difficult nonlinear ODE's?

Yes, there are various techniques such as perturbation methods, series solutions, or transformation methods that can be used to solve certain types of difficult nonlinear ODE's. However, these methods may not always provide an exact solution and may require additional assumptions or approximations.

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