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We are asked to solve the non-homogeneous linear difference equation: $$ Y_{t+2} + 15 Y_{t+1} + 5 Y_t = 1 + 6t + 13t^2 $$ Step 1: Solve the homogeneous equation The homogeneous equation is: $$ Y_{t+2} + 15 Y_{t+1} + 5 Y_t = 0 $$ Assume solution of the form $Y_t = r^t$. Then the characteristic equation is: $$ r^2 + 15 r + 5 = 0 $$ Solve for $r$ using the quadratic formula: $$ r = \frac{-15 \pm \sqrt{15^2 - 4 \cdot 1 \cdot 5}}{2} = \frac{-15 \pm \sqrt{225 - 20}}{2} = \frac{-15 \pm \sqrt{205}}{2} $$ So the roots are: $$ r_1 = \frac{-15 + \sqrt{205}}{2}, \quad r_2 = \frac{-15 - \sqrt{205}}{2} $$ Hence the homogeneous solution is: $$ Y_t^{(h)} = A \left(\frac{-15 + \sqrt{205}}{2}\right)^t + B \left(\frac{-15 - \sqrt{205}}{2}\right)^t $$ Step 2: Find a particular solution The non-homogeneous term is a quadratic _____ ________ __________ _______ ___ __________ _________ _____ ________.
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