\( y ^ { \prime \prime } - 4 y ^ { \prime } + 9 y = 6 x ^ { 2 } \)

This second-order linear differential equation features constant coefficients and a polynomial non-homogeneous term. The characteristic equation \( r^2 - 4r + 9 = 0 \) yields complex roots, specifically \( r = 2 \pm 3i \). The complementary solution hence takes the form \( y_c = e^{2x}(C_1\cos(3x) + C_2\sin(3x)) \). To find a particular solution, one can use the method of undetermined coefficients or variation of parameters. As the non-homogeneous part is a polynomial, the particular solution \( y_p \) is often guessed as \( Ax^2 + Bx + C \). Solving for coefficients by substitution gives you an explicit form to add to the complementary solution for a complete solution! Happy solving!