Question

Determine whether the given signal is a solution to the difference equation. Then find the general solution to the difference equation. y_k = k^2: y_k + 2 + 8y_k + 1 - 9y_k = 20k + 12 the given signal is a solution to the difference equation? No Yes What is the general solution to the difference equation? y_k = 20k + 12 + c_1 k^2 + c_2(-9)^k y_k = k^2 + c_1(-9)^k + c_2 y_k = 20k + 12 + c_1(-9)^k + c_2 Since y_k = k^2 is not a particular solution, there is not enough information to determine the general solution.

Answer 1

The answer to this question can be defined as follows:

Explanation:

The given equation is:

`y_(k + 2) + 8y_(k +1) - 9y_(k) = 20k + 12..(1)`

put,

`y_k = k^2\\\\y_(k+2)=(k+2)^2\\\\y_(k+1)=(k+1)^2\\\\`

`(k+2)^2+8(K+1)^2-9k^2 = 20k+12\\\\=20k+12= 20K+12\\\\`

hence y_k=k^2 is its solution.

Now,

`\to y_(k+2)+ 8y_k + 1 - 9y_k = 20k + 12`

the symbol form is:

`(E^2+8E-9)_(yk)=20k+12`

`\to m^2+8m-9=0\\\\\to m^2+(9-1)m-9=0\\\\\to m^2+9m-m-9=0\\\\\to m(m+9)-(m+9)=0\\\\\to (m+9)(m-1)=0\\\\\to m=-9 \ \ \ \ \ \ \ m=1\\`

The general solution is:

`y_k = c_1(-9)^k + c_2(`1)^k\\\\y_k =c_1(-9)^k+c_2`

The complete solution is:

`y_k=(y_k)_c+(y_k)_y\\\\y_k= c_1(-9)^k+c_2+k^2`

The answer is option b:
`y_k = k^2 + c_1(-9)^k + c_2`

After solve the complete solution is:

`\bold{y_1=c_1(-9)^k+c_2+k^2.....}`