$ \text{ Put 3x +2 =} e^z ,z=log(3x+2) \\ $

$ 9D(D-1)y + 3(3 )Dy - 36y = 3\frac{(e^z - 2) ^2} {3} + 4\frac{(e^z - 2) ^2} {3} + 1\\ $

$ (9D^2- 9D + 9D - 36)y = \frac{1}{3}(e^{2z} - 1) \\ $

$ (9D^2 - 36)y = \frac{1}{3}(e^{2z} - 1) \\ $

$ 9(D^2 - 4)y = \frac{1}{3}(e^{2z} - 1) \\ $

$ (D^2 - 4)y = \frac{1}{27} (e^{2z} - 1) \\ $

$ \text{ The Auxiliary equation is } \\ $

$ D^2 - 4 =0 \\ $

$ D = \pm 2 \\ $

$ \text{ C.F. is } y_c = c_1e^{2z} + c_2e^{-2z} \\ $

$ \text{P.I = } \frac{1}{D^2 - 4}\frac{1}{27} (e^{2z} - 1 ) \\ $

$ =\frac{1}{27} \frac{1}{D^2 - 4} (e^{2z} - 1 ) \\ $

$ =\frac{1}{27} [ \frac{1}{D^2 - 4} e^{2z} - \frac{1}{D^2 - 4} e^{0z} ] \\ $

$ =\frac{1}{27} [ \frac{z}{2D} e^{2z} + \frac{1}{ 4} ] \\ $

$ = \frac{1}{27} [ \frac{z}{4} e^{2z} + \frac{1}{ 4} ] \\ $

$\therefore \text{ The complete solution is y = C.F. + P.I. } \\ $

$ \therefore y = c_1e^{2z} + c_2e^{-z} + \frac{1}{27} [ \frac{z}{4} e^{2z} + \frac{1}{ 4} ] \\ $

$ \therefore y = c_1(3x+2)^{2} + c_2(3x+2)^{-2} + \frac{1}{27} [ \frac{log(3x+2)}{4} (3x+2)^{2} + \frac{1}{ 4} ] \\ $