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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 $a_{n}(0)$?

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

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