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I am solving an Ode using Mathematica: $$-p\alpha x^{p-1}=m\ddot{x},\quad x(0)=x_0,\quad x'(0)=x''(0)=0,\quad x(t_1)=0.$$ And I have $p\alpha>0$, all $x$ and $t$ are positive. I want to figure out $t_1$.

This is my code:

DSolve[
  {(-p α/m) x[t]^(p - 1) == x''[t],
   x[0] == x0, x'[0] == 0, x''[0] == 0, x[t1] == 0},
  x[t], t, 
  Assumptions -> p α > 0 && p > 0 && m > 0 && t1 > 0 && x0 > 0
]

And the error message is

DSolve: General solution contains implicit solutions. In the boundary value problem, these solutions will be ignored, so some of the solutions will be lost.

How can I do?

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  • $\begingroup$ For second order diff equation DSolve can solve with only 2 initial/boundary condition. $\endgroup$ Mar 19, 2023 at 14:02
  • $\begingroup$ Thank you! So which 2 of the initial conditions should I choose? $\endgroup$
    – dcmpsr
    Mar 19, 2023 at 14:06
  • 2
    $\begingroup$ x''[0] is already given by the equation and can not be specified independently. But your problem is the term x[t]^(p - 1). Depending on p you can get many widely different solutions. You should set the exponent to a fixed value. $\endgroup$ Mar 19, 2023 at 14:09
  • $\begingroup$ But p is an important factor in this problem. I’m afraid I can’t make it fixed. $\endgroup$
    – dcmpsr
    Mar 20, 2023 at 6:19

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