Maze1125 said:
renegade7 said:
But qualitative analysis isn't an explicit solution.
That not relevant, a solution is a solution.
Lets say that I need to show that x is less than c.
If I can show that x is less than or equal to s (where s is the maximum possible value of x) and that s is less than c then I have solved the problem without ever even needing to find the exact value of x.
The question of whether or not it's an "explicit" solution is meaningless, the problem has been solved.
When you're talking about explicitly solving a system of differential equations, that means more than just extracting the needed information for a particular application. Certainly not knocking on that approach: with most equations that's all you'll ever have to go on. Equations that can be fully solved are the exception, not the rule. Solving the problem of determining whether an object can levitate due to the combined gravitational forces of all objects in the solar system is possible by qualitative analysis, but it's not a solution to the system of equations.
An "explicit" solution to a differential equation, or a system of equations, is one that results in the equation no longer being differential, ie, you've eliminated all of the derivatives and integrals and have a function that directly relates the dependent variable to the independent variable(s). So the solution to the system of differential equations where the position x of an object at any time t is given directly by some function of the position of the Sun and planets is what an "explicit solution" would mean here, and it is such a solution that is impossible with n-body problems.