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Re: DSolve test

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  • Subject: [mg132728] Re: DSolve test
  • From: Roland Franzius <roland.franzius at uos.de>
  • Date: Wed, 14 May 2014 05:25:18 -0400 (EDT)
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Am 09.05.2014 08:08, schrieb carlos.felippa%colorado.edu at gtempaccount.com:
> Hi, could somebody try this in Mathematica 9 to see if the bug is fixed?
>
> sol = DSolve[{u'[t] == 2*Sqrt[u[t]], u[0] == 0}, u[t], t]; Print[sol];

In all versions until now, Mathematica  does not check the Lipshitz 
condition  |f(u(t))-f(u(0))| < C |u(t)-u(0)| that guaranties uniqueness.

The results for nonlinear ordinary differential equations are just those 
you find in the usual lists like Kamke oder EquationWorld.

There exist no general algebraic nonlinear solving methods for nonlinear 
ODE's except linear substitutions, separation of variables and lookup 
tables.

Generally as a student of ODE one learns to look for points of 
discontinuities, here u=0, which generally allow branching:

f(t):=0 /;t<=t0
f(t):=(t-t0)^2/; t>t0>=0

is the solution family on (-oo,oo).

Dsolve has no entry for a domain construct like NDSolve

  NDSolve[{u'[t] == 2*Sqrt[u[t]], u[0] == 0}, u[t],{t,0,10}]

that will give you the missing constant solution and. But

In[20]:= v[t_] =
  u[t] /. NDSolve[{u'[t] == 2 Sqrt[u[t]], u[1] == 10^-12},
     u[t], {t, 0, 7}][[1]]

There occurs an error in the internal procedure at the critical point 1:

 >>During evaluation of In[20]:= NDSolve::mxst: Maximum number of 10000 
 >>steps reached at the point t == 0.9998803430478553`.

So even with CASystems, it is still the fact that differentiation is an 
algebraic functor, numerical integration is a trivialtity for smooth 
functions, algebraic integration is a mystery and a historical source of 
many branches of mathematics.

The solution of nonlinear differential equations remains an ingenious 
kind of art, Ricatti, Clairaut, d'Alembert are some of the protagonists.

-- 

Roland Franzius



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