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:

$x\,dy = (y + 2x^3)\,dx$

Dividing both sides by $x\,dx$:

$\dfrac{dy}{dx} = \dfrac{y + 2x^3}{x} = \dfrac{y}{x} + 2x^2$

$\dfrac{dy}{dx} - \dfrac{y}{x} = 2x^2$

This is a first-order linear ODE of the form $\dfrac{dy}{dx} + P(x)\,y = Q(x)$, where $P(x) = -\dfrac{1}{x}$ and $Q(x) = 2x^2$.

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(A) A homogeneous DE requires $\dfrac{dy}{dx} = f\!\left(\dfrac{y}{x}\right)$, a function of $y/x$ only.

Here, $\dfrac{dy}{dx} = \dfrac{y}{x} + 2x^2$

The term $2x^2$ cannot be expressed as a function of $\dfrac{y}{x}$ alone.

(A) is FALSE.

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(B) Order $= 1$, since the highest derivative is $\dfrac{dy}{dx}$.

Degree $= 1$, since the power of $\dfrac{dy}{dx}$ is $1$.

Product $= 1 \times 1 = 1$

(B) is TRUE.

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(C) For a linear ODE, the integrating factor is $e^{\int P(x)\,dx}$.

Here $P(x) = -\dfrac{1}{x}$:

$\text{I.F.} = e^{\int -\frac{1}{x}\,dx}$

$= e^{-\ln x}$

$= \dfrac{1}{x}$

The integrating factor is $\dfrac{1}{x}$, not $x$.

(C) is FALSE.

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(D) Multiplying $\dfrac{dy}{dx} - \dfrac{y}{x} = 2x^2$ by the I.F. $= \dfrac{1}{x}$:

$\dfrac{1}{x}\dfrac{dy}{dx} - \dfrac{y}{x^2} = 2x$

The left side is the derivative of $\dfrac{y}{x}$:

$\dfrac{d}{dx}\left(\dfrac{y}{x}\right) = 2x$

Integrating both sides:

$\dfrac{y}{x} = \int 2x\,dx = x^2 + C$

$y = x^3 + Cx$

(D) is TRUE.

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(B) and (D) are TRUE.