I'm trying to solve the following differential equation, but it seems Mathematica doesn't give me an answer for that. Do you guys have any clue on how to make this work?

DSolve[{y''[x] == Sinh[y[x]]}, y[x], x]


  • 1
    $\begingroup$ What version are you using? 11.2 returns a result in terms of the Jacobi amplitude. $\endgroup$ Apr 17, 2020 at 16:13
  • 2
    $\begingroup$ same as JM's. On V 12.1 it works screen shot !Mathematica graphics but just gives warning about inverse function which is common $$\left\{\left\{y(x)\to -2 i \text{am}\left(\frac{1}{2} \sqrt{(-c_1-2) (x+c_2){}^2}|\frac{4}{c_1+2}\right)\right\},\left\{y(x)\to 2 i \text{am}\left(\frac{1}{2} \sqrt{(-c_1-2) (x+c_2){}^2}|\frac{4}{c_1+2}\right)\right\}\right\}$$ and always it is best to give version number when asking such questions. Mathematica changes from one version to another. $\endgroup$
    – Nasser
    Apr 17, 2020 at 16:15
  • 1
    $\begingroup$ Related Get a symbolic solution from DSolve $\endgroup$
    – Artes
    Apr 17, 2020 at 17:04

1 Answer 1


From the ordinary differential equation given in the question, it is easily seen that y==0 is a solution for all x. But that is too not the solution shown by Mathematica or Nasser.

input and output

The message is not an error message. It shows that the solution has a periodicity of some kind. This is the pure function type of solution so implicitly Mathematica already complains about missing boundary conditions. That avoids both the message and the representation as a pure function for the solution.

Have a look at the documentation of DSolve. Even more complicated at JacobiAmplitude.

A really nice help in Mathematica is:


This provides the mathematical terms and some of the boundary conditions that led to solutions. So the term is a family of solutions figured by Nasser even if y[0] and y'[0] are given.

enter image description here


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