\displaystyle
e^y+x^2y-e^{2x}=0\\
e^{y(x)}+x^2y(x)-e^{2x}=0\\
\\
e^{y(x)}y'(x)+2xy(x)+x^2y'(x)-2e^{2x}=0\\
\longrightarrow y'(x)=\frac{2e^{2x}-2xy(x)}{e^{y(x)}+x^2}\\
e^{y(x)}y'(x)y'(x)+e^{y(x)}y''(x)+2y(x)+2xy'(x)+2xy'(x)+x^2y''(x)-4e^{2x}=0\\
\longrightarrow y''(x)=\frac{4e^{2x}-2y(x)-4xy'(x)-e^{y(x)}y'(x)^2}{e^{y(x)}+x^2}\\
\\
y'(0)=\frac{2e^{0}}{e^{y(0)}}=\frac{2}{e^{y(0)}}\\
\\
e^{y(x)}+x^2y(x)-e^{2x}=0|_{x=0}\\
e^{y(0)}-1=0 \Rightarrow y(0)=0\\
\LongRightarrow y'(0)=2\\
y''(0)=\frac{4e^{0}-2y(0)-e^02^2}{e^{0}}=0\\