Help me with this math problem?

This is about finding the power series solution to a differential equation using power series substitution. I only know what's y, y' and y". After I substitute them into the differential equation, I don't know how to get to all those steps below. I got stuck on the fourth line of the problem. Can someone please help me with this and tell me how to sum up and shift the indices. Thank you.

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  • ted s
    Lv 7
    6 months ago

    1st : in the underlined equation are you dividing by x² ?

    2nd: in the ' highlighted' equation....you start with

    Σ { n = 4,..} n(n-1) a_n x^(n-2)....let w = n-3 ===> w = 1,2... &

     w + 3 = n....Σ {w = 1...} (w+3)(w+2) a_(w+3) x^(w+1) is the new sum

    now replace the symbol w  with the symbol n....you note the powers for x are the same , the integers used are the same { 1,2...}....thus you can now write the 3 sums as 1 series whose coefficients of x must take the value of 0....yielding a recurrence relationship of  a_(n+3) with a_n

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