HW1

1. Derive the Euler equation. Solve

min_{u(x),b} ∫_{0}^{b} ((u′)^{2} + 2x u) dx u(0) = 0, u(b) = 0,

2. Derive the Euler equation and boundary conditions. Solve. Find the first integral

min_{u(x)} [∫_{0}^{1} ((u′)^{2} − ω^{2}u^{2}) dx + 2u(1), ] u(0) = 0,

3. Derive the Euler equation and boundary conditions. Solve. Explain the dependence of the solution on a.

min_{u(x)} ∫_{0}^{b} ((u′)^{2} + a u u′ + u) dx u(0) = 0.

4. Derive the Euler equation and boundary conditions. Solve, using Maple

min_{u(x)} ∫_{0}^{b} ((u′′)^{2} − a (u′)^{2} + b u) dx u(0) = 0, u′(0) = 0