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Find a function that satisfies a given integral equation

by RoRi

Consider the equation

    \[ (f(x))^2 = \int_0^x f(t) \frac{\sin t}{2 + \cos t} \, dt. \]

Find a function f that is not zero everywhere and that is continuous for all x \in \mathbb{R} that satisfies this equation.

First, we take the derivative of both sides of the given equation (using the fundamental theorem of calculus),

    \begin{align*} && (f(x))^2 &= \int_0^x f(t) \frac{\sin t}{2+\cos t} \, dt \\[9pt] \implies &&2f(x) f'(x) &= f(x) \frac{\sin x}{2 + \cos x} \\[9pt] \implies &&2 f'(x) &= \frac{\sin x}{2+\cos x} \\[9pt] \implies &&f'(x) &= \frac{\sin x}{4+2 \cos x}. \end{align*}

Now we can integrate both sides,

    \begin{align*} f(x) &= \frac{1}{2} \int_0^x \frac{\sin t}{2+\cos t} \, dt \\[9pt] &= \left. -\frac{1}{2} (\log (2+\cos t)) \right|_0^x \\[9pt] &= -\frac{1}{2} (\log (2+\cos x) - \log 3)\\[9pt] &= \frac{1}{2} (\log 3 - \log (2+\cos x))\\[9pt] &= \log \sqrt{ \frac{3}{2+cos x}}. \end{align*}

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