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[?] Include x in the DE returned by the EulerEquations?

Daniel Voloshin

Daniel Voloshin, BMSTU

Posted 7 years ago

So, I need to find an extremal of a given functional J=Integrate[12xy+(y')^2,{x,1,3}] To do that I write the Euler's equation and get the DE: 12xy-2*y"=0. Then I try to do the same thing using Mathematica and EulerEquations[] function. And it returns just -2y"=0, ignoring 12x. What am I doing wrong? How to make Mathematica consider the independent variable?

POSTED BY: Daniel Voloshin

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Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 7 years ago

Syntax !!! Needs["VariationalMethods`"]; sol = EulerEquations[12xy[x] + y'[x]^2, y[x], x] (* 12 x - 2 (y^??)[x] == 0 ) First@DSolve[sol, y[x], x] ( {y[x] -> x^3 + C[1] + x C[2]} ) First@DSolve[{sol, y[1] == 0, y[3] == 26}, y[x], x] ( {y[x] -> -1 + x^3} *)

Syntax !!!

   Needs["VariationalMethods`"];
   sol = EulerEquations[12*x*y[x] + y'[x]^2, y[x], x]
   (* 12 x - 2 (y^??)[x] == 0 *)

   First@DSolve[sol, y[x], x]
   (* {y[x] -> x^3 + C[1] + x C[2]} *)

   First@DSolve[{sol, y[1] == 0, y[3] == 26}, y[x], x]
   (*  {y[x] -> -1 + x^3} *)

POSTED BY: Mariusz Iwaniuk