PDE. please only answer if youre sure about your solutions. incorrect or irrelevant answers will be downvoted.

5. (2 points for correct Euler-Lagrange equation and 3 points for correct boundary/initial condition) Derive the Euler-Lagrange equation, and its boundary/initial condi- tions, for the functional I(u) = 6" SL(t, t, u, a, u.,)dadt, where u has given fixed-end boundary conditions at boundary positions 3 = +0,1 and given initial condition at time t = 0. Do we need to prescribe a terminal condition for u at time t = T? Explain. 5. (2 points for correct Euler-Lagrange equation and 3 points for correct boundary/initial condition) Derive the Euler-Lagrange equation, and its boundary/initial condi- tions, for the functional I(u) = 6" SL(t, t, u, a, u.,)dadt, where u has given fixed-end boundary conditions at boundary positions 3 = +0,1 and given initial condition at time t = 0. Do we need to prescribe a terminal condition for u at time t = T? Explain