Re: DSolve validation

*To*: mathgroup at smc.vnet.net*Subject*: [mg34007] Re: [mg33989] DSolve validation*From*: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>*Date*: Fri, 26 Apr 2002 03:27:32 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Forgot one thing. A simpler way to do the verifying is: In[1]:= ode1 = Derivative[2][y][x] + x*y[x] == 0; In[2]:= sol1 = DSolve[ode1, y, x] Out[2]= {{y -> Function[{x}, AiryAi[(-1)^(1/3)*x]*C[1] + AiryBi[(-1)^(1/3)*x]*C[2]]}} In[3]:= FullSimplify[ode1 /. sol1] Out[3]= {True} On Thursday, April 25, 2002, at 10:06 PM, Andrzej Kozlowski wrote: > First of all, you forgot something. > > > In[1]:= > ode1 = Derivative[2][y][x] + x*y[x] == 0; > > In[2]:= > sol1 = DSolve[ode1, y[x], x] > > Out[2]= > {{y[x] -> AiryAi[(-1)^(1/3)*x]*C[1] + AiryBi[(-1)^(1/3)*x]* > C[2]}} > > In[3]:= > FullSimplify[ode1 /. D[sol1, x, x] /. D[sol1, x] /. sol1] > > Out[3]= > {{{True}}}} > > But actually this method will only work in relatively simple cases. > When the answer is complicated enough there is no algorithm to verify > it symbolically because there is no algorithm which reduces one > complicated expression to another. So you have to resort to trying to > obtain numerical confirmation of the validity of your solutions. > Andrzej Kozlowski Toyama International University JAPAN http://platon.c.u-tokyo.ac.jp/andrzej/ > > On Thursday, April 25, 2002, at 04:00 PM, Vladimir Bondarenko wrote: > >> Gentlemen, >> >> It's easy to double check the DSolve output here: >> >> In[1] := ode1 = y''[x] + x y'[x] == 0; >> >> In[2] := sol1 = DSolve[ode1, y[x], x] >> Out[2] = {{y[x] -> C[2] + Sqrt[Pi/2]*C[1]*Erf[x/Sqrt[2]]}} >> >> In[3] := ode1 /. D[sol1, x, x] /. D[sol1, x] >> Out[3] = {{True}} >> >> Life is great! Inspirited with the success, let's consider this ODE. >> >> In[4] := ode1 = y''[x] + x y[x] == 0; >> >> In[5] := sol1 = DSolve[ode1, y[x], x] >> Out[5] = {{y[x] -> AiryAi[(-1)^(1/3)*x]*C[1] + AiryBi >> [(-1)^(1/3)*x]*C[2]}} >> >> In[6] := ode1 /. D[sol1, x, x] /. D[sol1, x] >> Out[6] = {{-(x*AiryAi[(-1)^(1/3)*x]*C[1]) - x*AiryBi >> [(-1)^(1/3)*x]*C[2] + x*y[x] == 0}} >> >> >> (* Oops! The trick does not work ;-( *) >> >> In[7] := ode1 /. D[sol1, x, x] /. D[sol1, x]//FullSimplify >> Out[7] = {{x*(AiryAi[(-1)^(1/3)*x]*C[1] + AiryBi >> [(-1)^(1/3)*x]*C[2] - y[x]) == 0}} >> >> >> (* Not great, again *) >> >> In[8] := ode1 /. D[sol1, x, x] /. D[sol1, x] // ComplexExpand // >> FullSimplify >> Out[8] = {{x*(AiryAi[(-1)^(1/3)*x]*C[1] + AiryBi >> [(-1)^(1/3)*x]*C[2] - y[x]) == 0}} >> >> (* etc etc etc *) >> >> >> 'Fraid, the same double check trouble holds for the hundreds ODEs I >> have tried 8-( >> >> What might be a more or less streamlined way to validate the DSolve >> solutions? >> Any module (publicly) available? Any (including halp-baked & raw) >> ideas & hints? >> >> Thank you a lot for your time and efforts in advance. >> >> >> Sincerely, >> >> Vladimir Bondarenko >> >> >> >> >> >