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Solution of 4t dy/dt + y = t^3

The question is attached (along with the answer). It's asking me to guess the solution of this differential equation. I think I would have to use the reduction of order method if I were given a particular solution but I am not given one. I'm assuming there is a logical way to guess this rather than just randomly guessing in a brute force manner. Could someone explain to me what the thought process is for this?

Thanks in advance!

Re: Solution of 4t dy/dt + y = t^3