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

Answer:

The answer to this question can be defined as follows:

Step-by-step 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:

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