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Re: Apparent bug in 4.2 version DSolve ?

• To: mathgroup at smc.vnet.net
• Subject: [mg40706] Re: [mg40688] Apparent bug in 4.2 version DSolve ?
• From: Dr Bob <majort at cox-internet.com>
• Date: Mon, 14 Apr 2003 04:05:29 -0400 (EDT)
• References: <200304130619.CAA27394@smc.vnet.net>
• Reply-to: majort at cox-internet.com
• Sender: owner-wri-mathgroup at wolfram.com

Using Version 4.2.1, I get:

eqn = D[y[z, v], {z, 2}] + (2 v + 1 - z*z) y[z, v] == 0;
y[z, v] /. DSolve[eqn, y, {z, v}]

{(HermiteH[v, z]*C[1][v])/ E^(z^2/2) + (Hypergeometric1F1[-(v/2), 1/2, 
z^2]*C[2][v])/
    E^(z^2/2)}

Note that C[i][v] denotes an arbitrary function of v.  Initial conditions 
are needed to determine C[1][v] and C[2][v].

I suppose this is the Wronskian:

Flatten@(List @@@ soln) /. {C[a_][v] -> 1};
Factor[Det@{%, D[%, z]}]

(-2*v*(z*HermiteH[v, z]*Hypergeometric1F1[
      1 - v/2, 3/2, z^2] + HermiteH[-1 + v, z]*Hypergeometric1F1[
      -(v/2), 1/2, z^2]))/E^z^2

That's zero if v is 0, for instance.  I don't see where your expression for 
the Wronskian comes from.  Mathematica is not assuming v is an integer, by 
the way -- even, odd, or whatever.

Here's your second differential equation:

eqn2 = y''[z] + (2 (2 k) + 1 - z*z) y[z] == 0;
y[z] /. DSolve[eqn2, y, z]

{(C[1]*HermiteH[2*k, z])/E^(z^2/2) + (C[2]*Hypergeometric1F1[-k, 1/2, z^2]) 
/
    E^(z^2/2)}

If we solve that with v in place of 2k (or substitute v/2 for k in the 
solution), we get:

eqn3 = y''[z] + (2 v + 1 - z*z) y[z] == 0;
y[z] /. DSolve[eqn3, y, z]
y[z] /. DSolve[eqn2, y, z] /. k -> v/2

{(C[1]*HermiteH[v, z])/E^(z^2/2) + (C[2]*Hypergeometric1F1[-(v/2), 1/2, 
z^2])/
    E^(z^2/2)}
{(C[1]*HermiteH[v, z])/E^(z^2/2) + (C[2]*Hypergeometric1F1[-(v/2), 1/2, 
z^2])/
    E^(z^2/2)}

and that's a special case of the original solution for y[z,v].

Bobby

On Sun, 13 Apr 2003 02:19:39 -0400 (EDT), Jim Diamond <jimd at linfield.edu> 
wrote:

> I have been trying to use a notebook I generated under an earier
> version of Mathematica (3.0.0.0), and I have run into a difficulty:
> for the differential equation of interest
> y''[z,v] + (2 v + 1 - z*z) y[z,v] ==0
> the routine DSolve produces two solutions
>
> (a) y1[z,v] = E^(-z^2 /2) HermiteH[v,z]
> and
> (b)  y2[z,v]= E^(-z^2 /2) Hypergeometric1F1[-v/2,1/2,z^2]
>
> The Wronksian of these two solution is W = 2^v Sqrt[Pi] v / Gamma[1 - 
> v/2]
> which vanishes when v is an even positive integer, so these
> two solutions are in fact linearly dependent when v is an even
> positive integer.
>
> And of course the Hermite polynomials HermiteH[v,z] are even functions
> of z when v is an even integer, so it is clear that the general
> solution prouced by DSolve for an arbitrary parameter v does not
> include the solution of odd parity
> (c) y3[z,v] = E^(-z^2 /2) Hypergeometric1F1[(1-v)/2,3/2,z^2]
> The Wronskian of (b) and (c) is W = 1
>
> When one solves the differential equation
> y''[z] + (2 (2 k) + 1 - z*z) y[z] ==0
> where k is an integer,
> then one gets the two linearly independent solutions (b) and (c).
>
> But isn't the whole point of having an analytic general solution avoiding 
> having to obtain explicit solutions to a differential
> equation each time the parameters change?
>
> I wrote to Wolfram about this issue last week.
>
> Hardware:    AMD Athlon Processor 1,572,340 kB RAM
> OS:         Windows 2000 5.00.2195 Service Pack 3
> also  Windows 98 SE 4.10.2222 A
> Mathematica: 4.2.1.0 (Windows)
>
> I am troubled by this error.
>
> Sincerely,
> Jim
> --
> Jim Diamond
> Linfield College Chemistry Department
> McMinnville, OR 97128
>
>



-- 
majort at cox-internet.com
Bobby R. Treat

• References:
  • Apparent bug in 4.2 version DSolve ?
    • From: jimd@linfield.edu (Jim Diamond)