Question

y=f(x) thỏa mãn f'(x).f(x)=x\(^4\)+x\(^2\). biết f(0) =2. tính f\(^{^{ }2}\)(2) = ?

Answer 1

\(f'\left(x\right).f\left(x\right)=x^4+x^2\)

Lấy nguyên hàm 2 vế:

\(\int f\left(x\right).f'\left(x\right)dx=\int\left(x^4+x^2\right)dx\)

\(\Leftrightarrow\int f\left(x\right)d\left(f\left(x\right)\right)=\int\left(x^4+x^2\right)dx\)

\(\Leftrightarrow\frac{f^2\left(x\right)}{2}=\frac{1}{5}x^5+\frac{1}{3}x^3+C\)

\(\Rightarrow f^2\left(x\right)=\frac{2}{5}x^5+\frac{2}{3}x^3+C\)

\(x=0\Rightarrow f^2\left(0\right)=C\Rightarrow C=4\Rightarrow f^2\left(x\right)=\frac{2}{5}x^5+\frac{2}{3}x^3+4\)

\(\Rightarrow f^2\left(2\right)=\frac{2}{5}.2^5+\frac{2}{3}x^3+4=\frac{332}{15}\)