In order to find the solutions of a homogeneous differential equation of the form $y'' + p(t)y' + q(t)y = 0$ we may use $r^2 + p(t)r + q(t) = 0$ which will give a two-part solution to the system.
Let's say we have a homogeneous third order ODE like $y''' -2y'' +y' = 0$. You may factor this as $r(r^2 - 2r + 1)$. Solving with the method given above you get a three-part solution: $y_{h} = C_{1} + C_{2}e^t + C_{3}te^{t}$. This is indeed a solution, but what happens when we want to plug-in a particular solution $y_{p}$ for $y''' -2y'' +y' = \dfrac{te^t}{(t+1)^2}$?
It's clear that we can use variation of parameters and solve easily IF we choose to ignore the $C_{1}$ part of the homogeneous solution. Then our Wronskian is 2x2 and plugging-in values we get the general solution $e^t(log(t+1)(t+1) + c1 -c2 + c2t )$. Wolfram Alpha also uses this method but it seems to me that by erasing part of the general solution we could be erasing more possible solutions.
As you know the general solution in this case is $$ y=y_h + y_p $$ where your $y_h$ is the part with the constants $C_1, C_2, C_3$ and the $y_p$ part is any particular solution found by any method.
As long as you find a particular solution your general solution is found. All you want from the particular solution is to satisfy your original equation.
So, if by dropping the constant you can find a particular solution, that is fine, go for it.