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Mathematics
Differential Equations
ODE with non-exact solution: closed-form, non-iterative approximations
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[QUOTE="Grelbr42, post: 6870356, member: 732225"] There is a method to produce a solution that may be useful for some purposes. It's called "solution generation." The basic idea is, you choose a form of the function g() such that you can solve the system exactly. You choose a form that is as reasonably close to the actual form of g() as you can and still solve the system. For example, if you had g(f(t)) = f(t), that is, g() is just the identity, then you can clearly solve the system. But maybe that is too drastically different from the g() in your case. So you start looking at forms of g() that are closer to the real form, but still give you solutions. For example, you could look for forms that let you use a Laplace transform to solve the resulting differential equation. Or any other form that you can solve the resulting differential equation. If you can find a situation that is not drastically far from the real g(), then there are a large variety of perturbative methods to solve the resultant system. Which one you choose will depend on the details of the system and the form of solution that will work for your purposes. And you will always have the concern of whether such a scheme converges to the correct answer. [/QUOTE]
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Differential Equations
ODE with non-exact solution: closed-form, non-iterative approximations
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