12.8. Determine all differentiable functions $F:(0 ;+\infty) \rightarrow \mathbb{R}$, for which: $F(1)=1$ and $F\left(\frac{1}{x}\right) F^{\prime}(x)=\frac{1}{x} \ln x$.
Solution. Let $F^{\prime}(x)=f(x)$. Then $\left(F\left(\frac{1}{x}\right)\right)^{\prime}=-\frac{1}{x^{2}} f\left(\frac{1}{x}\right)$.

Since $F\left(\frac{1}{x}\right) f(x)=\frac{1}{x} \ln x$, we obtain that $F(x) f\left(\frac{1}{x}\right)=-x \ln x$. Multiplying each part of the last equality by $-\frac{1}{x^{2}}$ we get $-\frac{1}{x^{2}} F(x) f\left(\frac{1}{x}\right)=\frac{1}{x} \ln x$ and, respectively,

$$
\left(F\left(\frac{1}{x}\right)\right)^{\prime} F(x)=-\frac{1}{x^{2}} f\left(\frac{1}{x}\right) F(x)=-\frac{1}{x^{2}}(-x) \ln x=\frac{1}{x} \ln x
$$

We obtain

$$
F\left(\frac{1}{x}\right) F^{\prime}(x)=f(x) F\left(\frac{1}{x}\right)=\frac{1}{x} \ln x
$$

Adding equations (1) and (2) we get $\left(F\left(\frac{1}{x}\right) F(x)\right)^{\prime}=\left(F\left(\frac{1}{x}\right)\right)^{\prime} F(x)+F\left(\frac{1}{x}\right) F^{\prime}(x)=\frac{2}{x} \ln x$, Then $F\left(\frac{1}{x}\right) F(x)=\ln ^{2} x+C_{1}, C_{1} \in \mathbb{R}$.

The condition $F(1)=1$ implies $C_{1}=1$ and

$$
F\left(\frac{1}{x}\right) F(x)=\ln ^{2} x+1
$$

From $F\left(\frac{1}{x}\right) f(x)=\frac{1}{x} \ln x \Rightarrow f(1)=0$ and $x=1$ is the only zero of the function $f$, and $F$ has no zeros.

Multiplying $F\left(\frac{1}{x}\right) f(x)=\frac{1}{x} \ln x$ by $F(x)$ we get $F(x) F\left(\frac{1}{x}\right) f(x)=\frac{1}{x} \ln x \cdot F(x)$, which implies $\left(\ln ^{2} x+1\right) f(x)=\frac{1}{x} \ln x F(x) \Leftrightarrow\left(\ln ^{2} x+1\right) F^{\prime}(x)=\frac{1}{x} \ln x F(x) \Leftrightarrow$

$$
\begin{gathered}
\Leftrightarrow \frac{F^{\prime}(x)}{F(x)}-\frac{\ln x}{x\left(\ln ^{2} x+1\right)}=0 \Leftrightarrow\left(\ln |F(x)|-\ln \sqrt{\ln ^{2} x+1}\right)^{\prime}=0 \Leftrightarrow\left(\ln \frac{|F(x)|}{\sqrt{\ln ^{2} x+1}}\right)^{\prime}=0 \\
\Leftrightarrow \frac{F(x)}{\sqrt{\ln ^{2} x+1}}=C_{2} \Leftrightarrow F(x)=C_{2} \sqrt{\ln ^{2} x+1}, C_{2} \in \mathbb{R}
\end{gathered}
$$

The condition $F(1)=1$ implies $C_{2}=1$ and respectively $F(x)=\sqrt{\ln ^{2} x+1}$.