Question

Calculus of Variations: Obtain the Lagrange's equation for the extremal of the following functionals, then solve the resulting differential equation (consult any DEs textbook) \[ \begin{array}{l} x 1 x 2 y 2-y y^{\prime}+\left(y^{\prime}\right) 2 d x \\ x 1 x 2 y^{\prime} 1+x 2 y^{\prime} d x \\ x 1 x 2 y 2+\left(y^{\prime}\right) 2+2 y e x d x \end{array} \]

          Calculus of Variations:
Obtain the Lagrange's equation for the extremal of the following functionals, then solve the resulting differential equation (consult any DEs textbook)
\[
\begin{array}{l}
x 1 x 2 y 2-y y^{\prime}+\left(y^{\prime}\right) 2 d x \\
x 1 x 2 y^{\prime} 1+x 2 y^{\prime} d x \\
x 1 x 2 y 2+\left(y^{\prime}\right) 2+2 y e x d x
\end{array}
\]