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MathGroup Archive 1995

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Re: DSolve solution checking

  • To: mathgroup at christensen.cybernetics.net
  • Subject: [mg1048] Re: DSolve solution checking
  • From: withoff (David Withoff)
  • Date: Wed, 10 May 1995 08:45:37 -0400
  • Organization: Wolfram Research, Inc.

In article <3oeuta$611 at news0.cybernetics.net> bronstei at inf.ethz.ch (Manuel Bronstein) writes:
>In article <3o9mua$b92 at news0.cybernetics.net>,
>Rich Klopp <Rich_Klopp at qm.sri.com> wrote:
>>I am having difficulties checking the solution of a 2nd order ODE obtained
>>using DSolve.  I define the equation leftside == 0 using the next 3 lines:
>>
>>r[z] := a z^2 + g
>>
>>leftside = (r[z]^2 + e) D[u[z],{z,2}] +
>>	D[r[z]^2,z] D[u[z],z];
>>
>>bndEq = (leftside == 0)
>>
>>4 a z (g + a z^2) u'[z] + (e + (g + a z^2)^2) u''[z] == 0
>>
>> [...
>>However, mathematica 2.2.1 on the PowerMac doesn't seem to know.
>>
>>% == 0 //Short
>>
>>{<<1>>} == 0
>>
>>Why can't Mathematica tell if it got the right answer?
>
>My reply properly won't get posted if I tell you what I think about it. If you
>do need a correct solution however, the general solution to your equation has
>the form
>
>                    /           dz
>  u(z) = C1  +  C2  | -----------------------
>                    |  2 4         2    2
>                    / a z  + 2a g z  + g  + e
>
>for arbitrary constants C1 and C2  (yes it can be checked with the system
>I used to get it).
>
>-----------------------------------------------------------------------------
>
>    ____________                    Manuel Bronstein
>   /      /    /   /                bronstein at inf.ethz.ch
>  /---   /    /___/                 Institute for Scientific Computation
> /      /    /   /                  ETH Zurich, Switzerland
> ----  /    /   /                   Tel: [+41] (1) 632-7474
>                                    Fax: [+41] (1) 632-1172
>
>        http://www.inf.ethz.ch/department/WR/html/people/bronstein.html
>
>-----------------------------------------------------------------------------

This worked ok in Mathematica Version 2.2  (and Version 2.1) when I tried
it.  What version of Mathematica were you using?  Did you load the
Calculus`DSolve` package?

Here is what happened when I tried this in Version 2.2:

In[1]:= << Calculus`DSolve`

In[2]:= r[z] := a z^2 + g

In[3]:= leftside = (r[z]^2 + e) D[u[z],{z,2}] +
               D[r[z]^2,z] D[u[z],z];

In[4]:= bndEq = (leftside == 0)

                      2                       2 2
Out[4]= 4 a z (g + a z ) u'[z] + (e + (g + a z ) ) u''[z] == 0

In[5]:= sol = DSolve[bndEq, u, z]

                                                   1
Out[5]= {{u -> (C[2] + C[1] Integrate[---------------------------, #1] & )}}
                                           2           2    2   4
                                      e + g  + 2 a g #1  + a  #1

which is the solution suggested above.  The result can be checked
by substituting it back into the original equation.

In[6]:= Simplify[bndEq /. sol]

Out[6]= {True}

In the version of Mathematica under development, you don't need to
load the Calculus`DSolve` package, and the integral is evaluated
in closed form.  Even with Version 2.2, though, the result looks fine.

Dave Withoff
Research and Development
Wolfram Research


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