For each of the following, find a path that extremizes the given integral.

a. \(f_{1}^{2}\left(y^{\prime 2}+2 y y^{\prime}+y^{2}\right) d y, y(1)=0, y(2)=1\).

b. \(f_{0}^{2} y^{2}\left(1-y^{\prime 2}\right) d y, y(0)=1, y(2)=2\).

c. \(f_{-1}^{1} 5 y^{\prime 2}+2 y y^{\prime} d y, y(-1)=1, y(1)=0\).