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Putnam
1990 Putnam
B1
B1
Part of
1990 Putnam
Problems
(1)
Integral and Derivative Equation
Source: Putnam 1990 B1
7/12/2013
Find all real-valued continuously differentiable functions
f
f
f
on the real line such that for all
x
x
x
,
(
f
(
x
)
)
2
=
∫
0
x
[
(
f
(
t
)
)
2
+
(
f
′
(
t
)
)
2
]
d
t
+
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.
(
f
(
x
)
)
2
=
∫
0
x
[
(
f
(
t
)
)
2
+
(
f
′
(
t
)
)
2
]
d
t
+
1990.
calculus
integration
derivative
function
algebra
functional equation
Putnam