\(\left(x+2\right)f\left(x\right)+\left(x+1\right)f'\left(x\right)=e^x\)
\(\Leftrightarrow e^x\left(x+2\right)f\left(x\right)+e^x\left(x+1\right)f'\left(x\right)=e^{2x}\)
\(\Leftrightarrow\left[e^x\left(x+1\right).f\left(x\right)\right]'=e^{2x}\)
Lấy nguyên hàm 2 vế:
\(\Leftrightarrow e^x\left(x+1\right).f\left(x\right)=\int e^{2x}dx=\frac{1}{2}e^{2x}+C\)
Do \(f\left(0\right)=\frac{1}{2}\Rightarrow e^0\left(0+1\right).f\left(0\right)=\frac{1}{2}e^0+C\Rightarrow C=0\)
\(\Rightarrow e^x\left(x+1\right)f\left(x\right)=\frac{1}{2}e^{2x}\Rightarrow f\left(x\right)=\frac{e^{2x}}{2e^x\left(x+1\right)}=\frac{e^x}{2\left(x+1\right)}\)
\(\Rightarrow f\left(2\right)=\frac{e^2}{6}\)
