finding the type of a function

Question

$f$ is a productive function. $f: (0,+ \infty) \to \mathbb R$ for which it applies $f(1)=2$ and $$x^2 f'(x) + x f(x)=1 \, \forall x>0.$$ Find the type of $f(x)$.

Answer 1

Hint

$$x^2 f'(x) + x f(x)=1$$ $$x f'(x) + f(x)=\frac 1 x$$ $$(xf(x))'=\frac 1 x$$

$$xf(x)=\ln(x)+K$$ $$f(1)=2 \implies K=2$$ $$\boxed{f(x)=\frac {\ln(x)+2}x}$$