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Integral and Derivative Equation

Source: Putnam 1990 B1

July 12, 2013
calculusintegrationderivativefunctionalgebrafunctional equationPutnam

Problem Statement

Find all real-valued continuously differentiable functions ff on the real line such that for all xx, (f(x))2=∫0x[(f(t))2+(f′(t))2] dt+1990. \left( f(x) \right)^2 = \displaystyle\int_0^x \left[ \left( f(t) \right)^2 + \left( f'(t) \right)^2 \right] \, \mathrm{d}t + 1990.
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