CoV18

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Find the Euler equations for J(y,z)=\int _{a}^{b}\left[y''z'+xyz''+z'''y^{2}\right]dx\,

For a functional of several functions, create a system of equations using the formula

L_{{y_{i}}}-{\frac  {d}{dx}}L_{{y_{i}^{\prime }}}+{\frac  {d^{2}}{dx^{2}}}L_{{y_{i}^{{\prime \prime }}}}-\ldots =0

So

{\begin{cases}L_{{y}}-{\frac  {d}{dx}}L_{{y^{\prime }}}+{\frac  {d^{2}}{dx^{2}}}L_{{y^{{\prime \prime }}}}=0\\L_{{z}}-{\frac  {d}{dx}}L_{{z^{\prime }}}+{\frac  {d^{2}}{dx^{2}}}L_{{z^{{\prime \prime }}}}-{\frac  {d^{3}}{dx^{3}}}L_{{z^{{\prime \prime \prime }}}}=0\end{cases}}
{\begin{cases}xz^{{\prime \prime }}+2z^{{\prime \prime \prime }}y-{\frac  {d}{dx}}(0)+{\frac  {d^{2}}{dx^{2}}}(z^{\prime })=0\\0-{\frac  {d}{dx}}(y^{{\prime \prime }})+{\frac  {d^{2}}{dx^{2}}}(xy)-{\frac  {d^{3}}{dx^{3}}}(y^{2})=0\end{cases}}
{\begin{cases}xz^{{\prime \prime }}+2z^{{\prime \prime \prime }}y+z^{{\prime \prime \prime }}=0\\-y^{{\prime \prime \prime }}+{\frac  {d}{dx}}(y+xy^{\prime })-{\frac  {d^{2}}{dx^{2}}}(2yy^{\prime })=0\end{cases}}
{\begin{cases}xz^{{\prime \prime }}+(2y+1)z^{{\prime \prime \prime }}=0\\-y^{{\prime \prime \prime }}+2y^{\prime }+xy^{{\prime \prime }}-{\frac  {d}{dx}}(2y^{{\prime 2}}+2yy^{{\prime \prime }})=0\end{cases}}
{\begin{cases}xz^{{\prime \prime }}+(2y+1)z^{{\prime \prime \prime }}=0\\-y^{{\prime \prime \prime }}+2y^{\prime }+xy^{{\prime \prime }}-6y^{{\prime }}y^{{\prime \prime }}-2yy^{{\prime \prime \prime }}=0\end{cases}}
{\begin{cases}xz^{{\prime \prime }}+(2y+1)z^{{\prime \prime \prime }}=0\\2y^{\prime }+xy^{{\prime \prime }}-6y^{{\prime }}y^{{\prime \prime }}-(2y+1)y^{{\prime \prime \prime }}=0\end{cases}}

Which are the Euler equations for the functional.


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