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I am looking for a way to solve this equation symbolically with Mathematica.

F[u_] := u^2/2 - u^3/3;

Solve[(Integrate[(1/Sqrt[F[rhou] - F[s]]), {s, qu1, ut}])^2 == 
  2*ulambda*t^2, ut]
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  • $\begingroup$ What are rhou, qu1, ut, ulambda, t^2? Are they constants, real, positive? Why are you using ulambda t^2 instead of one symbol? $\endgroup$
    – Artes
    Jan 14, 2021 at 0:14
  • $\begingroup$ I want to find the value of ut for give other all real positive parametes: rhou, qu1, ulambda, t^2. t is in between 10^(-4) and 0.5. Yes, we can use one symbol for ulambda t^2. $\endgroup$
    – mathfun
    Jan 14, 2021 at 0:50
  • $\begingroup$ The problem is the Integration part. I do not know a way to solve this since there is this integration. $\endgroup$
    – mathfun
    Jan 14, 2021 at 0:54
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    $\begingroup$ You haven't provided sufficient information what does not work. Setting e.g. Integrate[(1/Sqrt[F[2/3] - F[s]]), {s, 0, t}, Assumptions -> t > 0] this evaluates to elliptic functions. How to solve such a problem see e.g. Solving equations involving integrals. E.g. Integrate[(1/Sqrt[F[1] - F[s]]), {s, 0, t}, Assumptions -> 0 < t < 1] yields an expression involving logarithm. You should not work with so many symbolic constants without restricting them appropriately. $\endgroup$
    – Artes
    Jan 14, 2021 at 1:14
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    $\begingroup$ Mathematica can do the anti-derivative, i.e. the integral without limits. But since you have given it no information on the constants, it cannot determine if the function is continuous within the limits. Even if you assume you can safely apply the limits after integration, the resulting transcendental equation probably cannot be solved analytically. $\endgroup$
    – Bill Watts
    Jan 14, 2021 at 1:19

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