Consider the linear difference equation yn2 6yn1 9yn 0 and

Consider the linear difference equation y_n+2 6y_n+1 + 9y_n = 0, and show that the guess y_j = r^j only results in one value of r. Then show hat the guess y_j = jr^j is also a solution, and thus is the second solution.

Solution

Substituting the guess

yj=r^j gives

r^2-6r+9=0

(r-3)^2=0

Hence,r=3

So we get only one value of r,

We need to now show that if r^n is solution then ,nr^n is also solution. Substituting gives

(n+2)r^{n+2}-6(n+1)r^{n+1}+9nr^n

=n(r^{n+2}-6r^{n+1}+9r^n)+2r^{n+2}-6r^{n+1}

=n*0+2*3^{n+2}-6*3^{n+1}

=0

Hence proved

Hence, n3^n is second solution