Deduce a related series from an implicit differential equation

Write the right side as $\ln(e^x+1)-\ln2$. Differentiating gives $\sec^2y\,(y^\prime)^2+\tan y\,y^{\prime\prime}=e^x/(e^x+1)$. A second differentiation gives $2\sec^2y\tan y\,(y^\prime)^3+3\sec^2y\,y^\prime y^{\prime\prime}+\tan y\,y^{\prime\prime\prime}=e^x/(e^x+1)-e^{2x}/(e^x+1)^2$.

At zero, $y=\pi/4$. The original relation gives $y^\prime(0)=0$; the differentiated relations give $y^{\prime\prime}(0)=1/2$ and $y^{\prime\prime\prime}(0)=1/4$. Thus $y=\pi/4+x^2/4+x^3/24+O(x^4)$.

The original equation gives $(\cot y)\ln((e^x+1)/2)=y^\prime$. Differentiating the analytic series yields $y^\prime=x/2+x^2/8+O(x^3)$.