1.

Consider the difference equation x _{n+ 2} 3 x _{n +1}+2 x _n=0.

1. Show that the general solution to this equation is

   
   

   Now suppose that x ₀ 10 and x ₁=20. Then A ₁ and A ₂ must satisfy the system of equations

   
   

2. Solve for A ₁ and A ₂ and find the solution to the above initial value problem.

2.

Solve the following difference equations subject to the specified x values and sketch the solutions:

1. x _n ?5 x _n ?1+6 x _{n ? 2}=0; x ₀ =2, x ₁=5.

2. x _{n +1} 5 x _n+4 x _{n ?1}=0; x ₁=9, x ₂=33.

3. x _n ?x _{n ?2}=0; x ₁=3, x ₂=5.

4. x _{n +2} 2 x _{n +1}=0; x ₀=10.

5. x _{n +2}+ x _{n +1} 2 x _n=0; x ₀=6, x ₁=3.

6. x _{n +1}=3 x _n ; x ₁=12.

3.

1. ^[*]In Section 1.3 it was shown that the general solution to equations (16a,b) is (22) provided ? ₁ ? ? ₂. Show that if ? ₁= ? ₂= ? then the general solution is

   
   

2. Solve and graph the solutions to each of the following equations or systems

   1. x _{n +2} ? x _n=0,

   2. x _{n +2} 2 x _{n +1}+