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A function f(x) (with x > 0) satisfies integral from 0 to 1 of f(tx) dt = n f(x). Then f(x) is:
- f(x) = c * x^((1-n)/n)
- f(x) = c * x^(n/(n-1))
- f(x) = c * x^(1/n)
- f(x) = c * x^(1-n)
Correct answer: f(x) = c * x^((1-n)/n)
Solution
The substitution turns the equation into integral₀^x f(u) du = n*x*f(x); differentiating gives a separable ODE whose solution is a power law f(x) = c*x^((1-n)/n).
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