I am trying to solve this recurrence relation: 
$a\_n=a\_{n-1}+4n-3$
I was trying to solve it using characteristic equation. First I found homogeneous solution:

$a\_n-a\_{n-1}=0$

$x=1$

$a\_n=A\*1^n$

And then I tried to find particular solution, but I got stucked. I tried to increase recursion level and add up the equations to get rid off the 3. And then tried to find particular solution in form of $an+b$ and substitute in recursion but I got solutions $a=0, b=0 $. I'm confused, can I even solve it using this method. Can I find some other form for finding particular solution instead of $an+b$ without increasing recursion level. If not what are some techniques I can use to find explicit solution. I wouldn't like to guess the solution and then prove it by induction.