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Find the solution to the following non-homogeneous difference equation:

$u_{n+2}=2u_{n+1} + 3u_{n} + 3^{n} (6n + \frac{3}{2})$

The homogeneous part was pretty straight forward, but I was having trouble with the particular solution, can anyone help?

KingLogic
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Jerry McKenzie
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1 Answers1

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Hint: look for a particular solution of the form $u_n = 3^n (a n^2 + b n)$.

Robert Israel
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  • Is this true for all general cases of these problems? Why an^2 + bn? – Jerry McKenzie Nov 19 '20 at 22:31
  • In general, for a constant-coefficient linear recurrence where the right side is of the form $\lambda^n p(n)$, where $p$ is a polynomial of degree $d$, if $\lambda$ is not a root of the characteristic polynomial (so that $u_n = \lambda^n$ is not a solution of the homogeneous recurrence), then there will be a solution $\lambda^n q(n)$ where $q$ is a polynomial of degree $d$. If $\lambda$ is a simple root of the characteristic polynomial, $q$ will be a polynomial of degree $d+1$ with no constant term (that's the case in this example). – Robert Israel Nov 19 '20 at 23:18
  • If $\lambda$ is a root of multiplicity $m$, $q$ will be a polynomial of degree $d+m$ with no terms in $n^j$ for $j < m$. – Robert Israel Nov 19 '20 at 23:18