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Find the extremals of \int _{a}^{b}(y^{2}+y'^{2}+2ye^{x})\,dx\,.

The Euler-Lagrange equations are in general for the integrand functional L\,

{\frac  {\partial L}{\partial y}}-{\frac  {d}{dx}}{\frac  {\partial L}{\partial y'}}+{\frac  {d^{2}}{dx^{2}}}{\frac  {\partial L}{\partial y''}}-{\frac  {d^{3}}{dx^{3}}}{\frac  {\partial L}{\partial y'''}}+...+(-1)^{k}{\frac  {d^{k}}{dx^{k}}}{\frac  {\partial L}{\partial y^{{(k)}}}}=0\,.

So in this problem the EL-equation is

{\frac  {\partial L}{\partial y}}-{\frac  {d}{dx}}{\frac  {\partial L}{\partial y'}}=0\,.

2y+2e^{x}-{\frac  {d}{dx}}2y'=0\,

2y+2e^{x}-2y''=0\,

y''-y=e^{x}\,

The characteristic equation is m^{2}-1=0,m=\pm 1\,

y_{h}=c_{1}e^{x}+c_{2}e^{{-x}}\,

y_{p}=Axe^{x}\,

y_{p}'=Axe^{x}+Ae^{x}\,

y_{p}''=Axe^{x}+Ae^{x}+Ae^{x}\,

y_{p}''-y_{p}=2Ae^{x}=e^{x},A={\frac  {1}{2}}\,

y_{p}={\frac  {1}{2}}xe^{x}\,

y(x)=c_{1}e^{x}+c_{2}e^{{-x}}+{\frac  {1}{2}}xe^{x}\,

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