ghdoru11
  • ghdoru11
Find the extreme functions, y(x),z(x) for the functional F[y,z]= integrate from 0 to 1 [y^2+z^2+4(y')^2-16(z')^2] dx
Differential Equations
chestercat
  • chestercat
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IrishBoy123
  • IrishBoy123
\[\triangle\]
ghdoru11
  • ghdoru11
i don't understand
IrishBoy123
  • IrishBoy123
hi @ghdoru11 that silly sign means I am watching and interested i think you you want extreme functions y(x),z(x) for: \[F[y,z]= \int_{0}^{1} y^2+z^2+4(y')^2-16(z')^2 \ dx\] i could have a go but i think that might be really counter productive as i'd be guessing the relationship between x,y,z etc ie i don't recognise some of the finer lingo so i can, in the first instance, tag some people who know way better than i and some way you will get some help if that doesn't work out.... @Phi @SithsAndGiggles

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ghdoru11
  • ghdoru11
thank you for your time,i didn't know what that sign means
IrishBoy123
  • IrishBoy123
Michele_Laino
  • Michele_Laino
hint: here you have to write the Euler-Lagrange equation, for both functions y, and z, namely: \[\Large \begin{gathered} \frac{{\partial f}}{{\partial y}} - \frac{d}{{dx}}\frac{{\partial f}}{{\partial y'}} = 0 \hfill \\ \hfill \\ \frac{{\partial f}}{{\partial z}} - \frac{d}{{dx}}\frac{{\partial f}}{{\partial z'}} = 0 \hfill \\ \end{gathered} \] where: \[\Large f\left( {y,y'z,z'} \right) = {y^2} + {z^2} + 4{(y')^2} - 16{(z')^2}\] furthermore, we need to know the initial conditions for \( \large y(x), \; z(x) \) at \( \large x=0, x=1 \)
Michele_Laino
  • Michele_Laino
please solve both the Euler-Lagrange differential equations, for \( \large y(x), \; z(x) \) respectively
anonymous
  • anonymous
I'm not well-versed in functional calculus, but it sounds really interesting...
ghdoru11
  • ghdoru11
this is wrong. right?
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