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DSolve ignoring linear solutions

POSTED BY: Martin Castillo
3 Replies
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POSTED BY: Michael Rogers

For what it's worth, the computed solution has the missing solutions as limits of the generic solution:

Limit[
 DSolveValue[
  {y'[x] == y[x]/ z[x], z'[x] == z[x]/ y[x]}, {y[x], z[x]}, x
  ] /. C[2] -> a - 1/C[1], 
 C[1] -> 0]

(*  {a + x, a + x}  *)

This is fairly typical of generic symbolic-algebraic solutions to nonlinear equations.DSolve does some checking of limits of the generic solution (at least which with the IncludeSingularSolutions -> True option), but it does not do a thorough analysis.
POSTED BY: Michael Rogers

It is a shortcoming of the software, which only deals with the generic case.

DSolve[{y'[x] == y[x]/ z[x],
  z'[x] == z[x]/ y[x],
  y[0] == 1, z[0] == 1},
 {y, z}, x]

gives empty solution set, while this

DSolve[{y'[x] == y[x]/ z[x],
  z'[x] == z[x]/ y[x],
  y[1] == 1, z[1] == 1},
 {y, z}, x]

gives a funny Indeterminate output, when it should give y[x] == z[x] == x.

The option IncludeSingularSolutionsdoes not help in this case.

The linear solutions can be recovered for example this way:

eqs = {y'[x] == y[x]/ z[x],
   z'[x] == z[x]/ y[x]};
linearSolutions = {y -> Function[x, a  x + b],
   z -> Function[x, c  x + d]};
linearSolutions /. SolveAlways[eqs /.
   linearSolutions,
  x]
POSTED BY: Gianluca Gorni
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