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Mathematica Problem
*Subject*: Mathematica Problem
*From*: jack at sun.acs.udel.edu
*Date*: Mon, 19 Jun 89 14:51:49 -0500
*Apparently-to*: mathgroup-out at yoda.ncsa.uiuc.edu
Perhaps someone can help solve the following Mathematica problem:
Given y' = y,
solve using a series solution and prove that the result is E[x].
It is proving that the result is E[x] that I have not been able to do.
Here is a sample session:
In[1]:= Derivative[n_][y][x_]:=Derivative[n-1][y][x]
In[2]:= Series[y[x],{x,0,5}]
2 3 4 5
y[0] x y[0] x y[0] x y[0] x 6
Out[2]= y[0] + y[0] x + ------- + ------- + ------- + ------- + O[x]
2 6 24 120
In[3]:= seriesbyy0[x_,n_]:=Normal[Expand[Cancel[Series[y[x],{x,0,n}]/y[0]]]]
In[4]:= expseries[x_,n_]:=Sum[x^i/i!,{i,0,n}]
In[5]:= seriesbyy0[x,6]
2 3 4 5 6
x x x x x
Out[5]= 1 + x + -- + -- + -- + --- + ---
2 6 24 120 720
In[6]:= expseries[x,6]
2 3 4 5 6
x x x x x
Out[6]= 1 + x + -- + -- + -- + --- + ---
2 6 24 120 720
In[7]:= seriesbyy0[x,100]-expseries[x,100]
Out[7]= 0
As shown, I can prove the results are identical for a constant value of
the argument "n", but not for general n. I realize that the identity is
proven by visual inspection, but I would like to prove it by showing the
identity
seriesbyy0[x,n] == expseries[x,n]
evaluates to true. I have tried LogicalExpand, but to no avail.
Jack Seltzer
jack at sun.acs.udel.edu
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