Calculus of variations

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   For the functional $J[y] = \int_0^1 \left( (y')^{2} + 6\,y^{2} \right) dx$, evaluate $J[y]$ at the trial function $y(x) = 3\,x$.

2. Warm-up

   For the functional $J[y] = \int_0^1 \left( (y')^{2} + 9\,y^{2} \right) dx$, evaluate $J[y]$ at the trial function $y(x) = 5\,x$.

3. Warm-up

   For the functional $J[y] = \int_0^1 \left( (y')^{2} + 3\,y^{2} \right) dx$, evaluate $J[y]$ at the trial function $y(x) = 2\,x$.

Problems shown under their original terms: Generated (template, solver-computed).