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**Homework Statement:**What actually is the particular solution of an ODE?

**Relevant Equations:**x

Consider the differential equation ##y'' + 9y = 1/2 cos(3x)##, if we wish to solve this we should first solve the auxiliary equation ##m^2 + 9 = 0## giving us ##m=3i,-3i##, this corresponds to the complementary function ##Asin(3x) + Bcos(3x)##. Then because the complementary function contains the RHS of the differential equation our guess for the particular integral is no longer ##Csin(3x) + Dcos(3x)## but rather ##x(Csin(3x) + Dcos(3x))##. Then we will find the derivatives and substitute into our original equation to find that ##C=1/12## and ##D=0##. Now say we were given some intial conditions and found that ##A=B=1##, is the "particular solution" ##1/12xsin(3x)## or is the particular solution ##sin(3x) + cos(3x) +1/12xsin(3x) ##? My textbook sometimes references the PI + CF with the constants evaluated as the particular solution, whereas other times it references the PI as the particular solution. This makes me confused.