Question

[Minh Nguyệt]

Cho hàm số f(x) thỏa mãn f'(x) + 2x.f(x) = f(x).lnx với f(x)≠ 0, ∀x và f(1) =1. Khi đó \(\left|f\left(2\right)\right|\) bằng ?

Answer 1

[Nguyễn Việt Lâm]

\(\Leftrightarrow\dfrac{f'\left(x\right)}{f\left(x\right)}+2x=lnx\Rightarrow\dfrac{f'\left(x\right)}{f\left(x\right)}=lnx-2x\)

Lấy nguyên hàm 2 vế:

\(\Rightarrow\int\dfrac{f'\left(x\right)}{f\left(x\right)}dx=\int\left(lnx-2x\right)dx\)

\(\Rightarrow ln\left|f\left(x\right)\right|=x\left(lnx-1\right)-x^2+C\)

Thay \(x=1\)

\(\Rightarrow ln\left|f\left(1\right)\right|=-2+C\Rightarrow C=2\)

\(\Rightarrow ln\left|f\left(x\right)\right|=x\left(lnx-1\right)-x^2+2\)

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

\(\Rightarrow\left|f\left(2\right)\right|\)