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