CoV14

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Compute the first variation of J(y)=\int _{a}^{b}(y'^{2}+2y)dx\,

\delta J(y,h)\, ={\frac  {d}{d\varepsilon }}J(y+\varepsilon h)\left.\right|_{{\varepsilon =0}}
={\frac  {d}{d\varepsilon }}\int _{a}^{b}((y^{\prime }+\varepsilon h^{\prime })^{2}+2(y+\varepsilon h))\ dx\left.\right|_{{\varepsilon =0}}
={\frac  {d}{d\varepsilon }}\int _{a}^{b}(y^{{\prime 2}}+2\varepsilon y^{\prime }h^{\prime }+\varepsilon ^{2}h^{{\prime 2}}+2y+2\varepsilon h)\ dx\left.\right|_{{\varepsilon =0}}
=\int _{a}^{b}{\frac  {d}{d\varepsilon }}(y^{{\prime 2}}+2\varepsilon y^{\prime }h^{\prime }+\varepsilon ^{2}h^{{\prime 2}}+2y+2\varepsilon h)\ dx\left.\right|_{{\varepsilon =0}}
=\int _{a}^{b}(2y^{\prime }h^{\prime }+2\varepsilon h^{{\prime 2}}+2h)\ dx\left.\right|_{{\varepsilon =0}}
=\int _{a}^{b}(2y^{\prime }h^{\prime }+2h)\ dx


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