Consider the differential equation xdy = (y+2x^3)dx . Then which of t…

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

Dividing by \(x\,dx\):

    \(\displaystyle \frac{dy}{dx} = \frac{y + 2x^3}{x} = \frac{y}{x} + 2x^2\)

The term \(2x^2\) is not a function of \(\frac{y}{x}\), so the equation is not homogeneous. Rearranged, it is a linear equation \(\frac{dy}{dx} + Py = Q\):

    \(\displaystyle \frac{dy}{dx} - \frac{1}{x}\,y = 2x^2\)

    \(\displaystyle \text{I.F.} = e^{\int -\frac{1}{x}\,dx} = e^{-\log x} = \frac{1}{x}\)

    \(\displaystyle y \cdot \frac{1}{x} = \int 2x^2 \cdot \frac{1}{x}\,dx = x^2 + C\)

    \(y = x^3 + Cx\)

The highest derivative is \(\frac{dy}{dx}\), and it occurs to the power 1.

So the equation is not homogeneous, its order is 1 and its degree is 1, the integrating factor is \(\frac{1}{x}\), and the general solution is \(y = x^3 + Cx\).

Checking the statements

(A) It is a homogeneous differential equation. — Incorrect. \(\frac{dy}{dx} = \frac{y}{x} + 2x^2\), and \(2x^2\) is not a function of \(\frac{y}{x}\).

(B) Product of the order and degree of the differential equation is one. — Correct. \(1 \times 1 = 1\).

(C) Integrating factor is \(x\). — Incorrect. The integrating factor is \(\frac{1}{x}\).

(D) General solution of the differential equation is \(y=x^3+ Cx\), where C is an arbitrary constant. — Correct. \(\frac{y}{x} = x^2 + C\) gives \(y = x^3 + Cx\).