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My differential equations have been written as

eqns = {Subscript[x, 1]'[t] == \[Alpha]*Subscript[x, 1][t]*Subscript[x, 3][t] -
  \[Beta]*Subscript[x, 1][t]*Subscript[x, 2][t], 
Subscript[x, 2]'[t] == \[Gamma]*Subscript[x, 2][t] - \[Delta]*Subscript[x, 1][t]*Subscript[x, 2][t], 
Subscript[x, 3]'[t] == \[Epsilon]*Subscript[x, 1][t] - \[Lambda]*Subscript[x, 2][t],
Subscript[x, 1][0] == 147100 + 15000,
Subscript[x, 2][0] == 178000,
Subscript[x, 3][0] == 723586 + 554776};

sol = DSolve[eqns, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}, t]
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Mathematica cannot solve this equation, so it returns the input. It is possible (likely) that this equation doesn't have a solution in terms of simple functions. Try NDSolve for a numerical solution. –  Szabolcs Apr 14 at 15:41

1 Answer 1

This is not an answer. I am just providing a version of the OPs code in an easier-to-read and easier-to-work-with form.

eqns = 
  {x1'[t] == α*x1[t]*x3[t] - β*x1[t]*x2[t], 
   x2'[t] == γ*x2[t] - δ*x1[t]*x2[t], 
   x3'[t] == ϵ*x1[t] - λ*x2[t], 
   x1[0] == 147100 + 15000, 
   x2[0] == 178000, 
   x3[0] == 723586 + 554776};

sol = DSolve[eqns, {x1, x2, x3}, t]
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