Consider the differential equation $xdy - (y+2x^3)dx$. Then which of the following are TRUE?

(A) It is a homogeneous differential equation.
(B) Product of the order and degree of the differential equation in one.
(C) Integrating factor is $x$.
(D) General solution of the differential equation is $y=x^3+ Cx$, where C is an arbitary constant.

Choose the correct answer from the options given below:

(B) and (D) only

The correct answer is Option (3) → (B) and (D) only

$\text{Given: }x\frac{dy}{dx}=y+2x^{3}\;\Rightarrow\;\frac{dy}{dx}-\frac{1}{x}y=2x^{2}$

$\text{(A) Not homogeneous (term }2x^{2}\text{ prevents dependence only on }y/x)$

$\text{(B) Order }=1,\ \text{degree }=1\ \Rightarrow\ \text{product }=1\ (\text{True})$

$\text{(C) Integrating factor }=e^{\int(-1/x)\,dx}=e^{-\ln x}=\frac{1}{x}\neq x\ (\text{False})$

$\text{(D) With IF } \frac{1}{x}:\ \frac{d}{dx}\!\left(\frac{y}{x}\right)=2x\ \Rightarrow\ \frac{y}{x}=x^{2}+C$

$\Rightarrow\ y=x^{3}+Cx\ (\text{True})$

True statements: (B) and (D)