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Implicit solutions for a system of de's

• To: mathgroup at smc.vnet.net
• Subject: [mg28706] Implicit solutions for a system of de's
• From: "Arnold Seiken" <seikena at union.edu>
• Date: Thu, 10 May 2001 07:54:50 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

Dear Mathematica Mavens,

The system of "almost linear" differential equations
x' = y,  y' = -x + 2 x^3 has the implicit solution y^2+x^2-x^4=k.

This can be shown by evaluating
Dt[y^2 + x^2 - x^4 ]/.{Dt[x] :> y, Dt[y] :> -x + 2 x^3}//Simplify

But Mathematica seemingly can not find this solution using DSolve. If you
try

DSolve[{x'[t] == y[t], y'[t] == -x[t] + 2x[t]^3},
  {x'[t], y'[t]}, t]

you get explicit solutions which are not very useful. Is there a way to 
force Mathematica
to generate the much simpler implicit solution?

In the Help Browser (after typing in DSolve), it says

"DSolve sometimes gives implicit solutions in terms of Solve."

Exactly what does that sentence mean? What are "implicit solutions
in terms of Solve"?  Are there any examples of these creatures?

Any help appreciated.

Arnold Seiken