Unique solution for a difference ODE?

Any idea how to find general solution

$$a'_{n}(t)= (n+\alpha )a_{n}(t) + \beta a_{n+1}(t) + \gamma a_{n+2}(t)$$

for some coefficients $$\alpha, \beta, \gamma$$?, Where $$a'_{n}(t)=\frac{d}{dt} a_{n}(t)$$?

I know using ansatz of the form $$a_{n}(t)=A(t) B(t)^n$$ one can find $$A(t)$$ and $$B(t)$$ and hence a solution for $$a_{n}(t)$$, but is it the general solution?

Is there a unique solution for above ODE with initial condition $$a_{n}(0)$$?

• I would like to know what happens if $a_n(t) = A(t)B(t)^n$? – Rinmyaku Mar 10 at 7:43