[Solved] MA3475 Exam 1

This is an open book open notes exam but you are not allowed to access the Internet or ask anyone for help. The answers must be 100% your work. Show your work as unsupported answers may receive no credit.

1. (5 points) Find the extremal of the functional

2. (5 points) Consider the functional
   • (2 points) Find the Euler-Lagrange equation of the above functional.
   • (1 point) Find r such that y(x) = x^r solves your Euler-Lagrange equation.
   • (2 points) Find the extremal of J[y] satisfying the boundary conditions y(1) = 0,

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3. (5 points) Determine the extremal of the functional

where , are nonzero constants for each of the following boundary conditions:

• y(0) = 0, y(1) = 1.
• y(0) = 0, y(1) is free.
• y(0) and y(1) are both free.