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RE: DSolve solution validation
*To*: mathgroup at smc.vnet.net
*Subject*: [mg33949] RE: [mg33910] DSolve solution validation
*From*: "DrBob" <majort at cox-internet.com>
*Date*: Wed, 24 Apr 2002 01:22:02 -0400 (EDT)
*Reply-to*: <drbob at bigfoot.com>
*Sender*: owner-wri-mathgroup at wolfram.com
Consider this:
Clear[f,g]
f[z_]:=Exp[z]
g[z_]:=-f[-z]
f'[z]-Abs[f[z]]//InputForm
E^z - E^Re[z]
g'[z]-Abs[g[z]] //InputForm
E^(-z) - E^(-Re[z])
So you see, here are two solutions (f and g) to your differential
equation, if z is restricted to Real values. Unfortunately, DSolve
doesn't make that assumption, or can't simplify the solutions properly.
Bobby Treat
-----Original Message-----
From: Vladimir Bondarenko [mailto:vvb at mail.strace.net]
To: mathgroup at smc.vnet.net
Subject: [mg33949] [mg33910] DSolve solution validation
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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