Yikes. Indeed - being silly and not thinking through my complex numbers.
Thanks for the detailed response(s).
On Tuesday, June 28, 2016 at 1:07:51 PM UTC-4, Aaron Meurer wrote:
>
> In particular, tanh(x + I*pi/2) = coth(x) and coth(x + I*pi/2) =
> tanh(x). You can verify this with SymPy:
>
> In [22]: x = Symbol('x', real=True)
>
> In [23]: tanh(x + I*pi/2).expand(complex=True).simplify()
> Out[23]:
> 1
> ───────
> tanh(x)
>
> In [24]: 1/tanh(x + I*pi/2).expand(complex=True).simplify()
> Out[24]: tanh(x)
>
> But note that the arbitrary constant in the solution of your ODE is
> like coth(a*t + C1). So the solution represents both tanh() and
> coth(), depending on the value of C1.
>
> Aaron Meurer
>
>
> On Tue, Jun 28, 2016 at 12:59 PM, Aaron Meurer <[email protected]
> <javascript:>> wrote:
> > The reason this is happening is that both tanh(x) and coth(x) satisfy
> > the same differential equation, namely f'(x) = 1 - f(x)**2, because
> > d/dx tanh(x) = 1 - tanh(x)**2 and d/dx coth(x) = 1 - coth(x)**2. This
> > is your ODE (up to a change in variable).
> >
> > Aaron Meurer
> >
> > On Tue, Jun 28, 2016 at 12:54 PM, Aaron Meurer <[email protected]
> <javascript:>> wrote:
> >> According to checkodesol(paraChute, solution), the solution is correct.
> >>
> >> Aaron Meurer
> >>
> >> On Tue, Jun 28, 2016 at 2:13 AM, Dan Lewis <[email protected]
> <javascript:>> wrote:
> >>> Hi folks,
> >>>
> >>> Looks like sympy/dsolve produces an incorrect solution to an ODE:
> >>>
> >>> from sympy import *
> >>>
> >>> mass, g, b, t = symbols('mass g b t')
> >>>
> >>> v, x = symbols('v x', cls=Function)
> >>>
> >>> paraChute = mass*v(t).diff(t)-mass*g+b*(v(t))**2
> >>>
> >>> solution = dsolve(paraChute,v(t),hint='lie_group')
> >>>
> >>> solution
> >>>
> >>> Out[1]:
> >>>
> >>> Eq(v(t), sqrt(g)*sqrt(mass)/(sqrt(b)*tanh(sqrt(b)*sqrt(g)*(C1*mass +
> >>> t)/sqrt(mass))))
> >>>
> >>>
> >>> Sorry in advance if I'm doing something silly - the solution shouldn't
> be
> >>> 1/tanh. Should be tanh.
> >>>
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