Question

Cho em xin lời giải bài 3 và bài 4 với ạ

Answer 1

Chỉ thấy bài 5 với 6:

5.

\(f'\left(x\right)+2f\left(x\right)=0\Leftrightarrow f'\left(x\right)=-2f\left(x\right)\Leftrightarrow\dfrac{f'\left(x\right)}{f\left(x\right)}=-2\)

Lấy nguyên hàm 2 vế:

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

Thay \(x=1\Rightarrow0=-2+C\Rightarrow C=2\)

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

\(\Rightarrow f\left(-1\right)=e^4\)

Answer 2

6.

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

\(\Leftrightarrow x'.f\left(x\right)+x.f'\left(x\right)=2x+1\)

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

Lấy nguyên hàm 2 vế:

\(\int\left[x.f\left(x\right)\right]'dx=\int\left(2x+1\right)dx\)

\(\Rightarrow x.f\left(x\right)=x^2+x+C\)

Thay \(x=1\Rightarrow1.f\left(1\right)=1+1+C\Rightarrow C=1\)

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

\(\Rightarrow f\left(2\right)=\dfrac{7}{2}\)

Answer 3

Ủa sao đề khác rồi:

3.

\(f'\left(x\right)=-e^x.f^2\left(x\right)\Leftrightarrow\dfrac{f'\left(x\right)}{f^2\left(x\right)}=-e^x\)

Lấy nguyên hàm 2 vế:

\(\int\dfrac{f'\left(x\right)}{f^2\left(x\right)}dx=\int-e^xdx\)

\(\Rightarrow\dfrac{1}{f\left(x\right)}=e^x+C\)

Thay \(x=0\Rightarrow2=1+C\Rightarrow C=1\)

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

\(\Rightarrow f\left(ln2\right)=\dfrac{1}{3}\)

Answer 4

4.

\(e^x\left[f\left(x\right)-f'\left(x\right)\right]=2f^2\left(x\right)\)

\(\Leftrightarrow\dfrac{\left(e^x\right)'.f\left(x\right)-e^x.f'\left(x\right)}{f^2\left(x\right)}=2\)

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

Lấy nguyên hàm 2 vế:

\(\int\left[\dfrac{e^x}{f\left(x\right)}\right]'dx=\int2dx\)

\(\Rightarrow\dfrac{e^x}{f\left(x\right)}=2x+C\)

Thay \(x=0\Rightarrow3=C\Rightarrow\dfrac{e^x}{f\left(x\right)}=2x+3\)

\(\Rightarrow f\left(x\right)=\dfrac{e^x}{2x+3}\)

\(\Rightarrow f\left(1\right)=\dfrac{e}{5}\)