Question

The particular integral of $$ (x^2 D^2 + 4 x D + 2) y = x^2 $$ is:

• A. \( \frac{e^{2x}}{12} \)
• B. \( \frac{x}{12} \)
• C. \( \frac{x e^x}{12} \)
• D. \( \frac{x^2}{12} \)

Hint:

Try polynomial trial solution matching RHS for particular integral.

Answer 1

The Correct Option is D

Trial solution for RHS \( x^2 \) is \( y = A x^2 \). Calculate derivatives and substitute in LHS operator: \[ D y = \frac{dy}{dx} = 2 A x, \quad D^2 y = 2 A \] Substitute: \[ (x^2)(2 A) + 4x (2 A x) + 2 (A x^2) = 2 A x^2 + 8 A x^2 + 2 A x^2 = 12 A x^2 \] Set equal to RHS: \[ 12 A x^2 = x^2 \implies A = \frac{1}{12} \] Thus, \[ y = \frac{x^2}{12} \]