I'm trying to find a increasing postive function [itex]\phi (x) [/itex] that minimizes the following integral for x in [0, L]:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \int_0^L A \frac{ d ^2 \phi (x) } {dx^2}+ (B +C cos( \phi (x)) ^2 \mbox{d}x [/tex]

with A and B real positve numbers and

[itex]\phi (0) =0 [/itex]

[itex]\phi ' (L) =0 [/itex]

When I use the the Lagrange equations I get:

[tex] \phi '' (x) + D sin(\phi (x) ) + E sin(\phi (x) ) cos( \phi (x) ) = 0 [/tex]

with D and E a constant.

Is this correct?

Can I find a numerical solution for this nonlinear ODE?

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# Minimization with cos()^2

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