Verify by direct substitution that the given series is a sol

Verify by direct substitution that the given series is a solution of the differential equation: y = Sigma^infinity_n = 1 (-1)^n + 1/n x^n (x + 1)y\" + y\' = 0

Solution

y = x - (x^2/2) + (x^3/3) - (x^4/4) + . . .

y\' = 1 - x + (x^2) - (x^3) + (x^4) - . . .

y\'\' = -1 + 2x - 3x^2 + 4x^3 - . . .

and (x+1)y\'\'

= (-x + 2x^2 - 3x^3 + 4x^4 - ...) + (-1 +2x - 3x^2 + 4x^3 ...)

= -1 + x - x^2 + x^3 - ...

= - y\'

and so (x+1)y\'\' + y\' = - y\' + y\' = 0