RE: DSolveConstants question

*To*: mathgroup at smc.vnet.net*Subject*: [mg3751] RE: DSolveConstants question*From*: rich_klopp at qm.sri.com (Richard W. Klopp)*Date*: Sun, 14 Apr 1996 03:02:44 -0400*Organization*: SRI International*Sender*: owner-wri-mathgroup at wolfram.com

I previously posted the following on comp.soft-sys.math.mathematica and received but one response, for the (Module[{C}, C] &) syntax question. I then wrote to Wolfram Support and received a couple of answers to my more central question. I am sharing these with the group with the hope that they help someone else, too. --------------------------------------------- In solving a system of differential equations, I run into the problem of Mathematica using the same constant of integration for all the equations, which is an unwanted behavior in my case, but I can't figure out an elegant workaround. I'd like my integration constants to be indexed the same as my equations. See the following example. Define a system of 2 equations: In[1]:= feq = Table[(f[i])'[x] + p[i] f[i][x] + q[i], {i,1,2}] Out[1]= {q[1] + p[1] f[1][x] + (f[1])'[x], q[2] + p[2] f[2][x] + (f[2])'[x]} Solve the equations with the requirement that any constants of integration be named cnst1[j] and note that, as (unhappily) expected, Mathematica restarts the index j for cnst[j] with each invocation of DSolve. In[2]:= Table[DSolve[feq[[i]] == 0, f[i][x], x, DSolveConstants -> cnst1], {i,1,2}] Out[2]= cnst1[1] q[1] {{{f[1][x] -> -------- - ----}}, x p[1] p[1] E cnst1[1] q[2] {{f[2][x] -> -------- - ----}}} x p[2] p[2] E ...... What I'd really like is for something like the following to work. That way, my various integration constants, for example cnst3[i][1], would have the same indexing system as my equations. How can I do this? In[4]:= Table[DSolve[ feq[[i]] == 0, f[i][x], x, DSolveConstants -> cnst[i] ], {i,1,2}] DSolve::csym: Value of option DSolveConstants -> cnst[1] must be a symbol....... Out[4]= ............. Is there a way to immediately substitute C[i][1] for C$1[1], say, and not get lost in the indexing? I eventually want to do this with systems of 12 or more equations. Thanks for your help! Rich Klopp SRI International 415-859-6482 PS **I've received an answer to the following: just get rid of the parenthesis and ampersand pure function syntax** THE BOOK, p. 783, says that I can use the option DSolveConstants -> (Module[{C}, C]&) to have Mathematica not restart the index, but this does not seem to work. Unfortunately, the book has no DSolveConstants examples to show how I've screwed up the systax. Can someone set me straight? I also guess that even if this did work, it would not be what I want because if there were more than one constant per equation, then the ith equation would have the nth and (n+1)th constants, with i != n. In[3]:= Table[DSolve[feq[[i]] == 0, f[i][x], x, DSolveConstants -> (Module[{cnst2}, cnst2]&)], {i,1,2}] DSolve::csym: Value of option DSolveConstants -> Module[{cnst2}, cnst2] & must be a symbol....... Out[3]= ========================================================================== The following is from David Withoff at Wolfram: --------------------------------------------------- I'm not entirely sure that I understand what you want to do, but hopefully the guesses below will be close enough to what you want to be useful. Here is one possibility. In[1]:= feq = Table[(f[i])'[x] + p[i] f[i][x] + q[i], {i,1,2}] Out[1]= {q[1] + p[1] f[1][x] + (f[1])'[x], q[2] + p[2] f[2][x] + (f[2])'[x]} In[2]:= Table[Module[{c}, DSolve[feq[[i]] == 0, f[i][x], x, DSolveConstants -> c] /. c -> cnst[i]], {i, 1, 2}] q[1] cnst[1][1] Out[2]= {{{f[1][x] -> -(----) + ----------}}, p[1] x p[1] E q[2] cnst[2][1] > {{f[2][x] -> -(----) + ----------}}} p[2] x p[2] E Here is another possibility. In[3]:= ConstantFixer[p_] := cnst[i][p] In[4]:= Table[DSolve[ feq[[i]] == 0, f[i][x], x, DSolveConstants -> ConstantFixer], {i,1,2}] q[1] cnst[1][1] Out[4]= {{{f[1][x] -> -(----) + ----------}}, p[1] x p[1] E q[2] cnst[2][1] > {{f[2][x] -> -(----) + ----------}}} p[2] x p[2] E This second example has the awkward feature of requiring that the free variable i in the definition of ConstantFixer have the same meaning as the local variable i in Table[..., {i, 1, 2}]. In other words, this method requires that i doesn't have a value. You can protect yourself from a global value for i using Block: In[5]:= Block[{i}, Table[DSolve[ feq[[i]] == 0, f[i][x], x, DSolveConstants -> ConstantFixer], {i,1,2}] ] q[1] cnst[1][1] Out[5]= {{{f[1][x] -> -(----) + ----------}}, p[1] x p[1] E q[2] cnst[2][1] > {{f[2][x] -> -(----) + ----------}}} p[2] x p[2] E which will work even if i has a global value. The strategy here is to do whatever you have to do to sneak the value of the DSolveConstants option past the argument checker in DSolve, which will only allow the value of the DSolveConstants option to be a symbol. Except in contrived examples the rest of DSolve doesn't care what the value of the DSolveConstants option is, so if you can get the value past the code that checks arguments, everything will work fine I believe that the argument checker in DSolve has already been changed for the next release of Mathematica such that the example you mentioned will work. Dave Withoff Research and Development Wolfram Research -- Rich Klopp rich_klopp at qm.sri.com ==== [MESSAGE SEPARATOR] ====

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