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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 == 
  2ulambdat^2, ut]
creidhne
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mathfun
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  • What are rhou, qu1, ut, ulambda, t^2? Are they constants, real, positive? Why are you using ulambda t^2 instead of one symbol? – Artes Jan 14 '21 at 00:14
  • 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. – mathfun Jan 14 '21 at 00:50
  • The problem is the Integration part. I do not know a way to solve this since there is this integration. – mathfun Jan 14 '21 at 00:54
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    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. – Artes Jan 14 '21 at 01:14
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    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. – Bill Watts Jan 14 '21 at 01:19
  • The positive parameter: rhou will be provided so that the function 1/Sqrt[F[rhou] - F[s]] is defined and continuous. Also the other positive parameters: qu1, ut, ulambda, t are such that the function is continuous. Please NOTE THAT the objective is to solve for ut symbolically (using Mathematica). Thus all parameters should remain as variables. Thank you for your comments.. Please let me know if you have more questions or comments. – mathfun Jan 14 '21 at 01:49
  • 1/Sqrt[F[rhou] - F[s]] is defined and continuous almost everywhere. Examine the post I've linked formerly and e,g, this post. I would have provided an answer if you had asked a reliable question. Nevertheless now I'm just discouraged with this question which deserves a downvoting. – Artes Jan 14 '21 at 02:44

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