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

$ 6D^2 +17D + 12 = 0 \\ $

$D= \frac {-4}{3} , \frac {-3}{2} \\ $

$ \text{ C.F. is } y = c_1 e^{\frac {-4}{3}}x + c_2 e^{ \frac {-3}{2}x} \\ $

$ \text{P.I = } \frac{1}{6D^2 +17D + 12} (e^{ \frac {-3}{2}x} + 2^x )\\ $

$= \frac{1}{6D^2 +17D + 12} e^{ \frac {-3}{2}x} + \frac{1}{6D^2 +17D + 12} e^{xlog2} $

$= \frac{x}{12(\frac{-3}{2}) + 17}e^{ \frac {-3}{2}x} + \frac{1}{ 6(log2)^2 +17log2 + 12} e^{xlog2}\\ $

$= -xe^{ \frac {-3}{2}x} + \frac{2^x}{ 6(log2)^2 +17log 2+ 12} \\ $

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

$ \therefore y =c_1 e^{\frac {-4}{3}}x + c_2 e^{ \frac {-3}{2}x} -xe^{ \frac {-3}{2}x} + \frac{2^x}{ 6(log2)^2 +17log2 + 12} $