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dharr

Dr. David Harrington

8482 Reputation

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21 years, 33 days
University of Victoria
Professor or university staff
Victoria, British Columbia, Canada

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Contact Dr. David Harrington
I am a retired professor of chemistry at the University of Victoria, BC, Canada. My research areas are electrochemistry and surface science. I have been a user of Maple since about 1990.

MaplePrimes Activity


These are answers submitted by dharr

difftorec...

dharr 8482
August 24 2017
2 0

Assuming that your x should be t, then

> 

restart;
> 

EQ:=(3*t)*diff(y(t),t,t)+y(t)=0;
gfun:-diffeqtorec(EQ, y(t), c(n));

{c(n)+(3*n^2+3*n)*c(n+1), c(0) = 0, c(1) = _t[2]}

> 

 


 

Download powerseries.mw

 

Workbooks and page break error...

dharr 8482
July 16 2017
0 1

There is a problem having the page breaks in the file "The Lattice Package Users Guide.mw" - if you remove them then you can load it into a workbook.

I figured this out by assembling the workbook by hand, and it might be instructive to do this. I would install a package from the cloud ("Orbitals" would be my choice but I'm biased) and look at the workbook structure to figure things out.

To make it a package in the cloud just save the workbook to the cloud from the file menu. Then you can install it from the cloud.

The help page ?worksheet,cloud,creatingPackages is quite good on the details of the process

solve using 2nd BC...

dharr 8482
April 05 2017
2 3
ode:=diff(y(x),x$2)+lambda*y(x)=0;
bc:=y(0)=0;
sol:=rhs(dsolve({ode,bc},y(x)));
solve(eval(sol,x=L),lambda,allsolutions);

gives  Pi^2*_Z1~^2/L^2 and then about(_Z1) tells you it is an integer.

Answer for loopless case...

dharr 8482
April 05 2017
0 9

(Edit: Didn't see Carl's solution until after I submitted this.)

There are n*(n-1) = 64 {0,1} adjacency matrices with zeroes on the main diagonal (loopless graphs), so for this case, an indexing function for generating the matrices can help:
 

> 

restart;with(GraphTheory):with(ListTools):
> 

perm:='perm':
n:=3;
bits:=n*(n-1);
indexfn:=proc(i,j) local m;
         m:=(i-1)*n+(j-1);
         if i=j then 0 else perm[m-iquo(m,n+1)] end if;
         end proc:
Matrix(n,n,indexfn);
mlist:=table():
for i from 0 to 2^bits-1 do
   perm:=Bits:-Split(i,'bits'=bits);
   mlist[i]:=Matrix(n,n,indexfn);
end do:
mlist:=convert(mlist,list):
> 

 

n := 3

  

bits := 6

  

Matrix([[0, perm[1], perm[2]], [perm[3], 0, perm[4]], [perm[5], perm[6], 0]])

 (1)
> 

cats:=[ListTools:-Categorize((i,j)->IsIsomorphic(Digraph(i),Digraph(j)),mlist)];

cats := [[Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = 1, (3, 1) = 1, (3, 2) = 1, (3, 3) = 0})]]

 (2)
> 

nops(cats);

16

 (3)
> 

nops(Flatten(cats));

64

 (4)
> 

 

 

Download graphs.mw

Strategy...

dharr 8482
April 05 2017
0 0

If the bugs go away, and you generate a list of the digraphs, then you can categorize into the groups (equivalence classes) that are isomorphic using

[ListTools:-Categorize(IsIsomorphic,graphlist)]

matrix and graph generation...

dharr 8482
April 05 2017
0 0

I would do the generation of the matrices and graphs like this (for the 2x2 case)

> 

n:=2;
bits:=n^2;
mlist:=table():
for i from 0 to 2^bits-1 do
   mlist[i]:=Matrix(n,n,Bits:-Split(i,'bits'=bits));
end do:
mlist:=convert(mlist,list);

n := 2

  

bits := 4

  

mlist := [Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 1, (2, 1) = 0, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 1, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 1, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 1, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 1, (2, 1) = 1, (2, 2) = 0}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 0, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 0, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 0, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 1, (2, 1) = 0, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 1, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 1, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 0, (1, 2) = 1, (2, 1) = 1, (2, 2) = 1}), Matrix(2, 2, {(1, 1) = 1, (1, 2) = 1, (2, 1) = 1, (2, 2) = 1})]

 (1)
> 

graphs:=map(GraphTheory:-Digraph,mlist):
> 

 


 

Download graphs.mw

indexed procedures...

dharr 8482
March 04 2017
0 1

You can set up indices with procedures, but you stll need parentheses to show it is a procedure.

> 

q := proc () local inds; if not type(procname, 'indexed') then return 'procname(args)' end if; inds := op(procname); inds end proc;

proc () local inds; if not type(procname, 'indexed') then return 'procname(args)' end if; inds := op(procname); inds end proc

 (1)
> 

q(y)

q(y)

 (2)
> 

q[4, 5]()

4, 5

 (3)
> 

q[4, 5]

q[4, 5]

 (4)
> 

``

 

Flawed logic...

dharr 8482
February 17 2017
0 20

If r=0 then you correctly change i. But then you test r agoinst N+1, which fails, and then the else clause sets i back to r, which is zero.

Loop weight...

dharr 8482
February 17 2017
0 1

The Digraph command doesn't seem to allow loops, but adding a weight in a diagonal entry of the Adjacency matrix seems to be accepted, and the weight is given by DrawGraph near the vertex.


 

> 

with(GraphTheory):
> 

A := Matrix([[2,2,0],[0,0,0],[1,3,0]]);

A := Matrix(3, 3, {(1, 1) = 2, (1, 2) = 2, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 3, (3, 3) = 0})

 (1)
> 

G := Graph(A);

GRAPHLN(directed, weighted, [1, 2, 3], Array(%id = 18446744074240237686), `GRAPHLN/table/20`, Matrix(%id = 18446744074240237566))

 (2)
> 

DrawGraph(G);

 
> 

Digraph({[1,1]});

Error, (in GraphTheory:-Graph) invalid edge/arc: [1, 1]

  
> 

 


 

Download Loopgraph.mw

Make some assumptions...

dharr 8482
February 11 2017
0 1

If you add "assuming M>0, g>0;" then you get undefined. If your further choose a limit from the left or right you get -infinity or +infinity.

Give the list of the new variables...

dharr 8482
February 11 2017
0 16

Maple needs to know the new varables are u and v, not W and Q. Try:

Change(V, {x = u-W, y = v-Q}, [u, v]);

Another way...

dharr 8482
December 06 2016
0 2

if Row(A,2).Row(A,2)=0 then...

missing multiplication...

dharr 8482
December 05 2016
0 2

Another 2-D math problem. There needs to be a multiplication between the c^2 and the parenthesis. Then it easily plots, and fsolve gives the two solutions, which are obviously x=+/-R.

y(x)2.mw

Not unique...

dharr 8482
November 30 2016
1 4

The companion matrix can be used - for monic polynomials. The determinant is not unique and may not be the one you want.
 

> 

with(LinearAlgebra):
> 

p:=w^2 + u*w + v; #monic polynomial

u*w+w^2+v

 (1)
> 

w*IdentityMatrix(degree(p,w))-CompanionMatrix(p,w);

Matrix([[w, v], [-1, w+u]])

 (2)
> 

 


 

Download Companion.mw

There is no exact solution, only a close...

dharr 8482
November 02 2016
2 2

There is no exact solution to the accuracy that Maple usually works at. You need to find a close "near enough" solution, which means you can't use fsolve. The easiest way is to minimize a sum of squares as in the worksheet below.
 

> 

eqs := {(2^2/a^2+2^2/c^2-2*(2*2)*cos(x)/(a*c))/sin(x)^2 = 1/2.2393^2, (2^2/a^2+4^2/c^2+4*(2*2)*cos(x)/(a*c))/sin(x)^2 = 1/1.5968^2, (1^2/a^2+sin(x)^2/b^2+2^2/c^2+(2*2)*cos(x)/(a*c))/sin(x)^2 = 1/2.7896^2, 1/sin(x)^2*(2^2*sin(x)^2/b^2) = 1/2.8650^2};

{(4/a^2+4/c^2-8*cos(x)/(a*c))/sin(x)^2 = .1994230893, (4/a^2+16/c^2+16*cos(x)/(a*c))/sin(x)^2 = .3921922000, (1/a^2+sin(x)^2/b^2+4/c^2+4*cos(x)/(a*c))/sin(x)^2 = .1285038476, 4/b^2 = .1218290191}

 (1)

Move the sin(x)^2 over to the right hand side

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eqs2:=eqs*~sin(x)^2;

{4*sin(x)^2/b^2 = .1218290191*sin(x)^2, 4/a^2+4/c^2-8*cos(x)/(a*c) = .1994230893*sin(x)^2, 4/a^2+16/c^2+16*cos(x)/(a*c) = .3921922000*sin(x)^2, 1/a^2+sin(x)^2/b^2+4/c^2+4*cos(x)/(a*c) = .1285038476*sin(x)^2}

 (2)

Sunm of squares of differences to minimize

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S:=add((lhs(eq)-rhs(eq))^2,eq in eqs2);

(4*sin(x)^2/b^2-.1218290191*sin(x)^2)^2+(4/a^2+4/c^2-8*cos(x)/(a*c)-.1994230893*sin(x)^2)^2+(4/a^2+16/c^2+16*cos(x)/(a*c)-.3921922000*sin(x)^2)^2+(1/a^2+sin(x)^2/b^2+4/c^2+4*cos(x)/(a*c)-.1285038476*sin(x)^2)^2

 (3)

Minimise subject to some constraints

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Optimization:-Minimize(S,{a>=0,b>=0,c>=0,x>=0,x<=Pi});

[0.162642134569434096e-11, [a = HFloat(8.902766366460824), b = HFloat(5.7300076115735505), c = HFloat(6.416618564610261), x = HFloat(1.8419029174006967)]]

 (4)

Gives a close solution, but not the one you wanted (though b is right). Probably there are several close solutions; it will be hard to decide - perhaps there are several combinations of a and c and x that work

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