Question

Xét hàm số f(x) liên tục trên đoạn R, thỏa mãn đk (x+2)f(x)+(x+1)f'(x)=e^x và f(0)=1/2. tính f(2)

Answer 1

\(\left(x+2\right)f\left(x\right)+\left(x+1\right)f'\left(x\right)=e^x\)

\(\Leftrightarrow e^x\left(x+2\right)f\left(x\right)+e^x\left(x+1\right)f'\left(x\right)=e^{2x}\)

\(\Leftrightarrow\left[e^x\left(x+1\right).f\left(x\right)\right]'=e^{2x}\)

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 \(f\left(0\right)=\frac{1}{2}\Rightarrow e^0\left(0+1\right).f\left(0\right)=\frac{1}{2}e^0+C\Rightarrow C=0\)

\(\Rightarrow e^x\left(x+1\right)f\left(x\right)=\frac{1}{2}e^{2x}\Rightarrow f\left(x\right)=\frac{e^{2x}}{2e^x\left(x+1\right)}=\frac{e^x}{2\left(x+1\right)}\)

\(\Rightarrow f\left(2\right)=\frac{e^2}{6}\)