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Higher-order, nonlinear differential equation with Initial Values


I tried to solve for an non-Hookean spring's motion, but the output from Mathematica is weird. It seems that there is inverse functions involved.


DSolve[{x''[t] + x[t]^3 == 0}, x[t], t]
Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>
Out[1] = {{x[t] -> -I 2^(1/4) Sqrt[-(1/Sqrt[C[1]])] Sqrt[C[1]]
 JacobiSN[Sqrt[
 Sqrt[2] t^2 Sqrt[C[1]] + 2 Sqrt[2] t Sqrt[C[1]] C[2] + 
  Sqrt[2] Sqrt[C[1]] C[2]^2]/Sqrt[2], -1]}, {x[t] -> I 2^(1/4) Sqrt[-(1/Sqrt[C[1]])] Sqrt[C[1]] JacobiSN[Sqrt[Sqrt[2] t^2 Sqrt[C[1]] + 2 Sqrt[2] t Sqrt[C[1]] C[2] + 

  Sqrt[2] Sqrt[C[1]] C[2]^2]/Sqrt[2], -1]}}

If you try Reduce, Mathematica will then give you no output at all, which makes sense because the output is not an equality or inequality.


Also, I added initial values into DSolve, but I'm unable to obtain the answer.


In[1]:= DSolve[{x''[t] + x[t]^3 == 0, x[0] == d, x'[0] == 0}, x[t], t]
Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>
DSolve::bvfail: For some branches of the general solution, unable to solve the conditions. >>
DSolve::bvfail: For some branches of the general solution, unable to solve the conditions. >>
Out[1]= {}

Answer




Well, that was easy:


Block[{Simplify = FullSimplify},
  DSolve[{x''[t] + x[t]^3 == 0, x'[0] == 0, x[0] == x0}, x[t], t]
  ] // FullSimplify
(*
  {{x[t] -> x0 JacobiCD[(t x0)/Sqrt[2], -1]}}
*)

DSolve uses Simplify to check the solution, and Simplify is not up to the task. So I used Block to replace it with FullSimplify, which will reduce the DE to True after DSolve substitutes the solution. Perhaps the Method option could be used, but there are no clues to how to use it.


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