Question : Find solution of the difference equation
p(n)=-4p(n-1)+5p(n-2)
Solution :
Given homogeneous linear difference equation \[ p_0=2,\ p_1=4,\] \[ p_n=-{4p}_{n-1}+5p_{n-2},\ \ n\geq2 \] or \[p_n+4p_{n-1}-5p_{n-2}=0\] (1) Replacing n by n+2 and taking n = 0, 1, 2, 3, 4, 5,…. \[p_{n+2}+4p_{n+1}-5p_n=0 \] So it’s auxiliary equation is \[\lambda^2+4\lambda-5=0\] \[\lambda^2+5\lambda-\lambda-5=0\] \[\lambda\left(\lambda+5\right)-1\left(\lambda+5\right)=0\] \[\left(\lambda+5\right)\left(\lambda-1\right)=0\] It’s roots are \[\lambda_1=-5 , \lambda_2=1\] Hence the general solution of the difference equation \[p_n=c_1\left(-5\right)^n+c_21^n,\ n\geq0 \] (2) Substituting n =0 and using first condition p0=2 \[2=c_1+c_2 \] (3) Substituting n =1 and using second condition p1=4 \[4=-5c_1+c_2 \] (4) And solving these equations we get \[c_1=-\frac{3}{2}\ \ ,\ \ c_2=\frac{14}{6}=\frac{7}{2}\] So the particular solution of the difference equation is given as \[ p_n=-\frac{3}{2}\left(-5\right)^n+\frac{7}{2}1^n,\ n\geq0\] Which can be written as \[p_n=\frac{1}{2}\left(7-3\left(-5\right)^n\right)\]