Let $$y=f(x)$$ be the real valued function defined on the interval $$(0,\infty)$$, satisfying $$y(1)=0$$ and the differential equation

$$x\dfrac{dy}{dx}=y-x^3.$$

Then which of the following statements is (are) TRUE?

A

The function $$f$$ has a local minimum at $$x=\dfrac{1}{\sqrt{3}}$$

B

The function $$f$$ has a local maximum at $$x=\dfrac{1}{\sqrt{3}}$$

C

The function $$f$$ is increasing in the interval $$(1,\,2)$$

D

If $$g(x)=4x^3-5x^2+\dfrac{3}{2}x$$ for $$x>0$$, then the number of elements in the set $$\{x\in(0,\infty):\;f(x)=g(x)\}$$ is $$2$$