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Putnam 1939 A5

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August 20, 2021
Putnam

Problem Statement

Do either (1)(1) or (2)(2)
(1)(1) xx and yy are functions of t.t. Solve x′=x+y−3,y′=−2x+3y+1,x' = x + y - 3, y' = -2x + 3y + 1, given that x(0)=y(0)=0.x(0) = y(0) = 0.
(2)(2) A weightless rod is hinged at OO so that it can rotate without friction in a vertical plane. A mass mm is attached to the end of the rod A,A, which is balanced vertically above O.O. At time t=0,t = 0, the rod moves away from the vertical with negligible initial angular velocity. Prove that the mass first reaches the position under OO at t=(OAg)ln⁡(1+sqrt(2)).t = \sqrt{(\frac{OA}{g})} \ln{(1 + sqrt(2))}.
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