Recurring relations solving differential equations

Huumah · Jan 6, 2013

Can anyone explain why we obtain the part where i put the red underline.
I understand everything until then

Textbook page deriving a power series recurrence for y'' + x^2 y = 0 about 0, shifting indices so both sums use x^n and ending with the combined series equal to zero.

Printed textbook page with text and equations on power series coefficients, including the combined sum 2(1)a2x^0 + 3(2)a3x + sum from n=2 of [(n+2)(n+1)a(n+2) + a(n-2)]x^n = 0, and the recurrence a(n+2) = -a(n-2)/((n+2)(n+1)) for n = 2, 3.

LCKurtz · Jan 6, 2013

In the line above your red line, write the first sum as the two sums$$
\left(\sum_{n=0}^1 + \sum_{n=2}^\infty\right)(n+2)(n+1)a_{n+2}x^n$$
The red lined terms come from the first sum. The second sum starting with ##n=2## is combined with the other like sum.

Recurring relations solving differential equations

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