Respuesta :
Answer:
The answer to this question can be defined as follows:
Step-by-step explanation:
The given equation is:
[tex]y_{k + 2} + 8y_{k +1} - 9y_{k} = 20k + 12..(1)[/tex]
put,
[tex]y_k = k^2\\\\y_{k+2}=(k+2)^2\\\\y_{k+1}=(k+1)^2\\\\[/tex]
[tex](k+2)^2+8(K+1)^2-9k^2 = 20k+12\\\\=20k+12= 20K+12\\\\[/tex]
hence y_k=k^2 is its solution.
Now,
[tex]\to y_{k+2}+ 8y_k + 1 - 9y_k = 20k + 12[/tex]
the symbol form is:
[tex](E^2+8E-9)_{yk}=20k+12[/tex]
[tex]\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\\[/tex]
The general solution is:
[tex]y_k = c_1(-9)^k + c_2(`1)^k\\\\y_k =c_1(-9)^k+c_2[/tex]
The complete solution is:
[tex]y_k=(y_k)_c+(y_k)_y\\\\y_k= c_1(-9)^k+c_2+k^2[/tex]
The answer is option b: [tex]y_k = k^2 + c_1(-9)^k + c_2[/tex]
After solve the complete solution is:
[tex]\bold{y_1=c_1(-9)^k+c_2+k^2.....}[/tex]
