A sequence u is defined by the firstorder linear difference

A sequence u is defined by the first-order, linear difference equation

Solve the LDE to obtain a formula for u[k].

Steps would be great.

Solution

First we solve assocaited homogeneous recurrence

u[k+1]=3u[k]

We look for solutions of the form :u[k]=r^k

Substituting gives, r=3

So, u[k]=A*3^k

Now for particular solution

Let, u[k]=Bk+C

Substituting gives

B(k+1)+C=3(Bk+C)-5k

Bk+B+C=3BK+3C-5k

B=2C

2Bk=5k so B=5/2

C=5/4

u[k]=5k/2+5/4

HEnce general solution is

u[k]=A*3^k+5k/2+5/4

u[0]=6=A+5/4

A=19/4