$16$A Variational Principles

Derive the Euler$-$Lagrange equation for the integral

${\displaystyle \underset{a}{\overset{b}{}}}f(x,y,y^{\text{'}},y^{\text{'}\text{'}})dx$

where prime denotes differentiation with respect to $x,$ and both $y$ and $y^{\text{'}}$ are specified at $x=a,b.$

Find $y(x)$ that extremises the integral

${\displaystyle \underset{0}{\overset{}{}}}(y+\frac{1}{2}{y}^2-\frac{1}{2}{y}^{\text{'}\text{'}2})dx$

subject to $y(0)=-1,y^{\text{'}}(0)=0,y()=cosh$ and $y^{\text{'}}()=sinh.$

Show that your solution is a global maximum. You may use the result that

${\displaystyle \underset{0}{\overset{}{}}}{}^2(x)dx{\displaystyle \underset{0}{\overset{}{}}}{}^{\text{'}2}(x)dx$

for any $($suitably differentiable$)$ function $$ which satisfies $(0)=0$ and $()=0.$