Problem 1
Write out the characteristic equation for each of the following, and find the character. istic roots: (a) \(y_{t+2}-y_{t+1}+\frac{1}{2} y_{t}=2\) (b) \(y_{1+2}-4 y_{t+1}+4 y_{t}=7\) (c) \(y_{1+2}+\frac{1}{2} y_{t-1}-\frac{1}{2} y_{t}=5\) \(\left(d^{\prime}\right) y_{t+2}-2 y_{t+1}+3 y_{t}=4\)
Problem 2
Find the particular solution of each of the following: \((a) y_{t+2}+2 y_{t-1}+y_{t}=3^{t}\) (b) \(y_{t+2}-5 y_{t+1}-6 y_{t}=2(6)^{t}\) (c) \(3 y_{t+2}+9 y_{t}=3(4)^{t}\)
Problem 4
Solve the following difference equations: \((a) y_{t+2}+3 y_{t+1}-\frac{7}{4} y_{t}=9 \quad\left(y_{0}=6 ; y_{1}=3\right)\) (b) \(y_{t+2}-2 y_{t+1}+2 y_{t}=1 \quad\left(y_{0}=3 ; y_{1}=4\right)\) (c) \(y_{i+2}-y_{t+1}+\frac{1}{4} y_{t}=2 \quad\left(y_{0}=4 ; y_{1}=7\right)\)
Problem 6
Test the convergence of the solutions of the following difference equations by the Schur theorem: \((a) y_{t+2}+\frac{1}{2} y_{t+1}-\frac{1}{2} y_{t}=3\) \((b) y_{t+2}-\frac{1}{9} y_{t}=1\)
