# differential equations – Solving an ODE with paramters and taking the limit of the solution

I am very new to Mathematica and already spent a lot of time trying to do this but failed.

I am trying to solve an ODE:

``````solution = DSolve({-((m (1 + m) + 4/(9 (-2/3 + t) t)) y(t)) +
2 (-1/3 + t) y'(t) + (-2/3 + t) t y''(
t) == (-4 (1 + C/2))/(9 (-2/3 + t) t), y(1) == 1, y'(1) == C},
y, t)
``````

where $$m$$ is a nonnegative integer and $$C$$ is a real number. (I am not sure how to incorporate that $$m$$ is a nonnegative integer in the code. )

I want to show that there exists a $$C$$ such that the solution is $$0$$ at infinity.
When I try that code:

``````Limit(y(t) /. solution((1)), t -> Infinity, m (Element) Integers)
``````

it just spits out the same thing.

What should I do?
(Note that I don’t need to find that value of $$C$$; I just need to show that for every $$m$$, there is a number $$C$$ in which the solution vanishes at infinity. )