Question 1

(a) Show that the Euler-Lagrange equation for the functional \[ S[y]=\int_{0}^{\frac{\pi}{2}}\left(y^{\prime 2}-y^{2}+4 y e^{-x}\right) d x, \quad y(0)=0, y(\pi / 2)=1, \] is \[ y^{\prime \prime}+y=2 e^{-x} . \] (b) Show that the stationary function is given by \[ y=e^{-x}-\cos x+\left(1-e^{-\pi / 2}\right) \sin x . \]

Question 2:

(a) Use the first integral of the Euler-Lagrange equation to show that the functional \[ S[y]=\int_{0}^{1} \frac{y^{4}}{y^{5}} d x, \quad y(0)=1, y(1)=\frac{1}{16}, \] has two real-valued stationary paths, given by \[ y(x)=(x+1)^{-4} \text { and } y(x)=(3 x-1)^{-4} . \] (b) Find the value of the functional on each stationary path.

Public Answer

GHMV21 The First Answerer