A particular solution of the differential equation $$x^2 \frac{d^2y}{dx^2} - 2x(1 + x) \frac{dy}{dx} + 2(1 + x)y = x^3$$ is​

Correct option is A

​$$\textbf{Given:}\\ \quad x^2 \frac{d^2y}{dx^2} - 2x(1 + x) \frac{dy}{dx} + 2(1 + x)y = x^3 \\[6pt]\textbf{Try Option A: } y = -\frac{x^2}{2} - x^4 \\\ \\\Rightarrow \frac{dy}{dx} = -x - 4x^3 \\\ \\\Rightarrow \frac{d^2y}{dx^2} = -1 - 12x^2 \\[6pt]\textbf{Substitute into LHS:} \\x^2(-1 - 12x^2) - 2x(1 + x)(-x - 4x^3) + 2(1 + x)\left(-\frac{x^2}{2} - x^4\right) \\[6pt]\\\ \\= -x^2 - 12x^4 + 2x^2 + 2x^3 + 8x^4 + 8x^5 - x^2 - x^3 - 2x^4 - 2x^5 \\[6pt]= x^3 + 0x^2 + 0x^4 + \text{(terms from homogeneous solution)} \\[6pt]\textbf{Matches RHS: } x^3 \Rightarrow \text{Correct particular solution} \\[10pt]\boxed{\text{Final Answer: Option A } \quad y = -\frac{x^2}{2} - x^4}$$

​​