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I am wondering what the essential value of an implicit solution to an (exact) ODE is? In ODE books usually implicit solutions occur in the context of exact differential equations, e.g.

$e^x\sin(x)+e^y\cos(y)\frac{dy}{dx}=0$ is an exact equation and the solution is given implicitly through the equation $\frac{1}{2}(\sin(x)-\cos(x))e^x+\frac{1}{2}(\sin(y)+\cos(y))e^y.$

In this example the solution can not be made explicit as it is often the case. Hence books usually leave it at that. But what do I gain with such an implicit solution? Sometimes it takes even a lot of effort to find an appropriate integrating factor first and then a potential function.

Therefore I suppose there must be at least some qualitative properties of the implicitly given solution which can be obtained from the algebraic equation in contrast to the defining ODE?!

Maybe someone can recommend good lterature where such questions are discussed?

I would welcome any help! Best wishes

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I can give three loosely related answers. First, you can do pretty good predictor-corrector numerics on an implicit solution. Without knowing the function which is held constant, it is difficult to specify the corrector part, so you can wind up gradually deviating from the level set as you run your numerics if you are not careful.

Second, in a sense finding the function whose level sets make the trajectories is backwards from what we often want to do in physics. That is because the function whose level sets form the trajectories is conserved. In physical problems, often it is the total energy. We often know this in advance and want to go the other way to find the appropriate evolution equations for the underlying variables.

Third, studying the critical points of the conserved quantity enables you to draw a contour plot, which has similar functionality to a phase portrait.

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  • $\begingroup$ Thank you very much! Do you know maybe good literature where I can find more information? $\endgroup$
    – asd
    Commented Oct 28, 2015 at 9:13
  • $\begingroup$ @asd It really depends on which part you mean. For the first part I already gave the buzzword that you would use to find such methods. For the second part you would need to consult a physics reference of some sort. The last part can probably be found in the same sort of place that you would learn about phase portraits. $\endgroup$
    – Ian
    Commented Oct 28, 2015 at 14:51
  • $\begingroup$ Nice answer! Especially the relevance to physics. Knowing/identifying conserved properties is very important to physicists. $\endgroup$
    – Behnam Esmayli
    Commented Apr 30, 2018 at 20:33

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