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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.

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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.

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