The characteristic equation is

 r − 3 = 0

The solution is r = 3. The homogeneous recurrence has general solution

 a_n = c₁3ⁿ

A particular solution is

 a_n = c₂n + c₃

for some constants c₂ and c₃.

Since a₀ = 2 and a₁ = 3a₀ + 1 = 7,

 2 = a₀ = c₂(0) + c₃.
 7 = a₁ = c₂(1) + c₃.

The first equation tells us that c₃ = 2. Then the second equation tells us that 7 = c₂ + 2, or c₂ = 5. So the particular solution is

 a_n = 5n + 2.

Notice that a₂ = 3a₂ + 2 = 3(7) + 2 = 23. Our particular solution gives a₂ = 5(2) + 2 = 12, which is not correct. The problem is that we need to add a term corresponding to the solution to the homogeneous equation. We get

 a_n = 5n + 2 + c₁3ⁿ

for some constant c₁. Since a₂ = 23,

 23 = a₂ = 5(2) + 2 + c₁3²

telling us that c₁ = 11/9. The final solution is

 a_n = 5n + 2 + (11/9)(3ⁿ).