Question Number 218891 by Nicholas666 last updated on 17/Apr/25
$$ \\ $$$$\:\boldsymbol{{suppose}}\:\boldsymbol{{y}}\left(\boldsymbol{{x}}\right)\:=\:\:\underset{\boldsymbol{{n}}=\mathrm{0}} {\overset{\infty} {\sum}}\boldsymbol{{a}}_{\boldsymbol{{n}}} \boldsymbol{{x}}^{\boldsymbol{{n}}} \boldsymbol{{statisfies}}\: \\ $$$$\:\:\:\boldsymbol{{y}}''\boldsymbol{{y}}−\left(\boldsymbol{{y}}'\right)^{\mathrm{2}} =\boldsymbol{{e}}^{\boldsymbol{{y}}} −\mathrm{1}\boldsymbol{{with}}\:\boldsymbol{{y}}\left(\mathrm{0}\right)=\mathrm{0}\:\boldsymbol{{and}}\:\boldsymbol{{y}}'\left(\mathrm{0}\right)=\mathrm{1}.\:\:\:\:\: \\ $$$$\:\:\:\:\boldsymbol{{determin}}\:\boldsymbol{{a}}_{\mathrm{4}} . \\ $$$$ \\ $$