Grouping like terms,
dy/dx = 3y - 1/2x - 3
In general, we want a differential equation of this type to follow the form dy/dx + p(x)y = q(x).
dy/dx - 3y = -1/2x - 3
color(red)(p(x) = -3)
Multiply by the integrating factor mu(x) = exp int color(red)(p(x))dx.
mu(x) = exp int color(red)(-3)dx = color(blue)(e^{-3x})
color(blue)(e^{-3x})dy/dx - 3color(blue)(e^{-3x})y = (-1/2x - 3)color(blue)(e^{-3x})
Notice that the left hand side is the derivative of a product.
d/dx(color(blue)(e^{-3x})y) = (-1/2x - 3)e^{-3x}
Integrate both sides.
int d/dx(color(blue)(e^{-3x})y) dx = -int(1/2x + 3)e^{-3x}dx
e^{-3x}y = 1/3(1/2x + 3)e^{-3x} + 1/18e^-3x + C
y = 1/3(1/2x + 3) + 1/18 + Ce^{3x}
y = 1/6x + 19/18 + Ce^{3x}