Theoretical/non-tedious question about Frobenius method

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The discussion centers on the application of the Frobenius method for solving differential equations using power series solutions. The user questions the justification for isolating the indicial polynomial when the overall solution is expressed as a sum. The key conclusion is that the isolation of the indicial polynomial is justified by the principle of equating coefficients, which allows for the determination of the indicial roots necessary for the solution. This method is crucial for understanding the behavior of solutions near singular points.

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When using the Frobenius method of solving differential equations using power series solutions, I get a solution

y = (indicial_stuff) + (infinite_summation_stuff) = 0

for a differential equation differential_stuff = 0.

WHY is it that I can say

(indicial_stuff) = 0?

If

y = (indicial_stuff) * (infinite_summation_stuff) = 0

(NOTICE THE MULTIPLICATION INSTEAD OF ADDITION) then it would make sense to me that

(indicial_stuff) = 0

but with the addition, I cannot make sense of this.

I tried to generalize this question and hope that I haven't made it more confusing but, if I have made the question confusing, please tell me and I will clear things up as best as I can.

Basically, I'm just asking for the reason why this – the isolation of the indicial polynomial equation (and the other summation portion) - is justified.

Any input would be greatly appreciated!
Thanks in advance!
 
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The answer to my question was: "by equating coefficients."
 

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