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For my characteristic roots method for solving a homogeneous recurrence relation, I got the roots for the equation as $2,2,3,3$.

for satisfying the boundary conditions, will the general solution be

$2^n(c_1+c_2\cdot n)+3^n(c_3+c_4\cdot n)$?

Please guide.

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    $\begingroup$ assuming a degree four recurrence with constant coefficients, the set of solutions for the homogeneous equation is a vector space of dimension 4; you have given 4 and they are linearly independent (over the field of constants), so those are it. But you should check by hand that each of the four really works.... $\endgroup$ – Will Jagy Apr 20 '14 at 19:31
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This looks right - it would help clarity if you gave the actual recurrence you are working with - I assume it is $a_{n+4}=10a_{n+3}-37a_{n+2}+60a_{n+1}-36a_n$

Have you tried checking that your proposed solution works? Since four items determine the whole thing, you need the four coefficients, but since this is linear you could check $2^n, n2^n, 3^n$ and $n3^n$ separately.

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