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Find the functions!

Source: Science ON 2021 grade XII/1

March 16, 2021
functional equationintegrationdifferentiable functioncalculus

Problem Statement

Find all differentiable functions f,g:[0,∞)→Rf, g:[0,\infty) \to \mathbb{R} and the real constant k≥0k\geq 0 such that \begin{align*} f(x) &=k+ \int_0^x \frac{g(t)}{f(t)}dt \\ g(x) &= -k-\int_0^x f(t)g(t) dt \end{align*} and f(0)=k,f′(0)=−k2/3f(0)=k, f'(0)=-k^2/3 and also f(x)≠0f(x)\neq 0 for all x≥0x\geq 0.\\ \\ (Nora Gavrea)
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