Your generic form, y''+ p(t)y'+ q(t)y= g(t) has 1 as the coefficient of y''. Your given equation does not: the coefficient of y'' is t^2- 3t. To get it to the correct form, you need to divide through by t^2- 3t:

y''+ [t/(t^2+ 3t)] y'- [(t+ 3)/(t^2+ 3t)]y= 0. t^2+ 3t= t(t+ 3) so after reducing the fractions, that is y''+ [1/(t+3)]y'- [1/t]y= 0. p(t)= 1/(t+3) and q(t)= -1/t.