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If I am asked to solve a differential equation "analytically", is this necessarily asking for explicit closed/open form solutions? Or are there other "analytical" techniques out there? ie: Would investigating solutions with phase diagrams count as an analytical technique?

For example: $$\dot{v}=v-\frac{v^3}{3}-w+a$$

$$\dot{w}=v+b-cw$$

where $a,b,c$ are real constants.

We can construct phase diagrams to analyse the trajectories for various parameter values, but we can't obtain explicit solutions for it.

The issue here is that I am not too sure if the term "analytically" is to be understood to be subjective, or if it has a very specific definition in the context of differential equations.

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  • $\begingroup$ I think this is just a question of naming, but in many classical books phase portrait are considered as a qualitative technique :) "doesn't provide solutions, but describes the behaviour of trajectories" $\endgroup$ – Evgeny Sep 15 '16 at 7:08

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