CoV28

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Compute the first variation J(y)=\int _{0}^{1}3y^{2}+xdx+y^{2}(0),y_{0}(x)=x,h(x)=x+1\,

\delta J(y_{0},h)={\frac  {d}{d\epsilon }}\left[\int _{0}^{1}3(y_{0}(x)+\epsilon h(x))^{2}+xdx+(y_{0}(0)+\epsilon h(0))^{2}\right]{\Bigg |}_{{\epsilon =0}}\,

={\frac  {d}{d\epsilon }}\left[\int _{0}^{1}3(x+\epsilon (x+1))^{2}+xdx+(0+\epsilon )^{2}\right]{\Bigg |}_{{\epsilon =0}}\,

={\frac  {d}{d\epsilon }}\left[\int _{0}^{1}3x^{2}+6\epsilon x(x+1)+3\epsilon ^{2}(x+1)^{2}+xdx+\epsilon ^{2}\right]{\Bigg |}_{{\epsilon =0}}\,

={\frac  {d}{d\epsilon }}\left[7\epsilon ^{2}+5\epsilon +\epsilon /2+\epsilon ^{2}\right]{\Bigg |}_{{\epsilon =0}}\,

=14\epsilon +5+2\epsilon {\Big |}_{{\epsilon =0}}\,

=5\,


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