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DSolve solution validation
*To*: mathgroup at smc.vnet.net
*Subject*: [mg33910] DSolve solution validation
*From*: Vladimir Bondarenko <vvb at mail.strace.net>
*Date*: Tue, 23 Apr 2002 07:13:19 -0400 (EDT)
*Reply-to*: Vladimir Bondarenko <vvb at mail.strace.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Sometimes, it is easy to double check the DSolve's output.
However, an attempt to solve
DSolve[y'[z] == Abs[y[z]], y[z], z]
yields
{{y[z] -> InverseFunction[(-(Log[2*Sqrt[Im[K$3541]^2 + Re[K$3541]^2] +
(2*(-Im[K$3541]^2 + Im[K$3541]*Im[#1] - Re[K$3541]^2 + Re[K$3541]*Re[#1]))/
Sqrt[Im[K$3541]^2 - 2*Im[K$3541]*Im[#1] + Im[#1]^2 + Re[K$3541]^2 -
2*Re[K$3541]*Re[#1] + Re[#1]^2]]/Sqrt[Im[K$3541]^2 - 2*Im[K$3541]*Im[#1] +
Im[#1]^2 + Re[K$3541]^2 - 2*Re[K$3541]*Re[#1] + Re[#1]^2]) + Log[2*Sqrt[Im[#1]^2 +
Re[#1]^2] + (2*((-Im[K$3541])*Im[#1] + Im[#1]^2 - Re[K$3541]*Re[#1] + Re[#1]^2))/
Sqrt[Im[K$3541]^2 - 2*Im[K$3541]*Im[#1] + Im[#1]^2 + Re[K$3541]^2 -
2*Re[K$3541]*Re[#1] + Re[#1]^2]]/Sqrt[Im[K$3541]^2 - 2*Im[K$3541]*Im[#1] +Im[#1]^2 +
Re[K$3541]^2 - 2*Re[K$3541]*Re[#1] + Re[#1]^2])*(-K$3541 + #1) & ][z + C[1]]}}
If I try to use D[] to check it, I get an expression with ByteCount of
737608. The attempt to Simplify this huge expression for the derivative
gave no answer after 1 hour at CPU = Athlon FX 1600+ / RAM = 512 Mb.
Question # 1: Is the above shown solution correct? (It involves 2 free constants,
K$3541 and C[1], but after simplification K$3541 might disappear?
(Also, this ODE is not linear, so the answer to it might have 2 free
constants?)
By hand I found
z Sign[y[z]] - Log[y[z]] = C[1]
Question # 2: Is this implicit solution correct? (I am not 100% sure)
Question # 3: Looks like I've calculated a collection of ODEs about which
I suspect that the corresponding DSolve's solutions are invalid.
What are the possible ways to double check these solutions?
Any modules written in Mathematica? Anything else?
Vladimir Bondarenko
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