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ghdoru11
 one year ago
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')^216(z')^2] dx
ghdoru11
 one year ago
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')^216(z')^2] dx

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IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.0hi @ghdoru11 that silly sign means I am watching and interested i think you you want extreme functions y(x),z(x) for: \[F[y,z]= \int_{0}^{1} y^2+z^2+4(y')^216(z')^2 \ dx\] 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

ghdoru11
 one year ago
Best ResponseYou've already chosen the best response.0thank you for your time,i didn't know what that sign means

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1hint: here you have to write the EulerLagrange 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
 one year ago
Best ResponseYou've already chosen the best response.1please solve both the EulerLagrange differential equations, for \( \large y(x), \; z(x) \) respectively

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I'm not wellversed in functional calculus, but it sounds really interesting...

ghdoru11
 one year ago
Best ResponseYou've already chosen the best response.0this is wrong. right?
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