I apologize if this question is far too obvious, but I'm a bit confused (and new to this subject). I have to solve the recurrence relation:
Using Euler substitution, I get
Solving this equation gives me $x_1=x_2=3$
The solution seemed to be obvious (since there weren't any initial conditions, I cannot determine $\lambda$ coefficients): $a_n=\lambda_1*3^n$
But the official one from my book is: $a_n=\lambda_1*3^n+\lambda_2*n*3^n$
The only explanation I'm left with is "Along with $a_n=\lambda_1*3^n$, it's obvious that $\lambda_2*n*3^n$ is another solution as well, which gives us $a_n=\lambda_1*3^n+\lambda_2*n*3^n$.
If anyone understands how they got the second solution, I would really appreciate an explanation. I assume it's something really obvious, but I can't see it. If the question isn't clear, please point me to it since I'm not sure I've translated it properly.