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So I've been wondering about this for a long time. Of course, Mathematica DSolve can be a monster slayer sometimes but as Mathematica as it can get, the output results usually tend to be not simplified. But I'm not concerned with the solution itself rather, I am using DSolve for some complicated ODEs to see if analytic solutions exist. In certain cases, the output $y=y(x)$ solution has a term with an unsolved integral. What may I infer from this?

  1. No analytic solutions exist. Hence Mathematica refuses to evaluate the integral part.
  2. Mathematica overcomplicated certain expressions during its computation causing the integral to lie beyond its computational time, even though the ODE has a solution(s).

Any thoughtful, and proper insights into this are appreciated!

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    $\begingroup$ Will you show examples -- both your input Mathematica codes and the corresponding results? $\endgroup$
    – A. Kato
    Commented Jul 27 at 7:40
  • $\begingroup$ has a term with an unsolved integral. does it also have the word "Solve" in the solution? If so this is implicit solution as it could not solve the integral. But it would be easier if you just gave an example of what you mean so no guessing will be needed. $\endgroup$
    – Nasser
    Commented Jul 27 at 20:56
  • $\begingroup$ "No analytic solutions exist": I guess you mean in the sense of "integration in finite terms"? -- "... with an unsolved integral": You can Activate the unsolved integral and see what happens. If it takes longer than the original DSolve, then probably DSolve time-constrained it. -- Overall though, failure to find a solution is no guarantee that no solution exists. That's been shown several times on this site. $\endgroup$
    – Michael E2
    Commented Jul 28 at 2:33

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