Solve the Euler-Lagrange equation associated with each of the integral functions $I[y]={\int }_a^{b}f\left(x,y,\mathrm{y\text{'}}\right)dx$ given below:

$f=\sin \mathrm{y\text{'}}$ $a=0$ , $b=1$ , $y(0)=-5$ and $y(1)=2$

$f=y{}^2-\mathrm{y\text{'}}{}^2$ , $a=0$