Question

Use the power series method to solve the given initial-value problem. (Enter the first four nonzero terms.) (x + 1)y" - (2 - x)y' + y = 0, y(0) - 8, 7(0) = -1 4 3 5 4 + y = 8-x-5x? Need Help? Read it Watch It Talk to a Tutor Submit Answer

Answer 1

noi: The Taylor expansion of ylon) about Z y (0) г no .Ht Y(0) + yllo) / And DE y"co)ni't y "(0)3724 are given y (0)=8, y'(o)= - 1 the given is & (0+1) y"- (2-) y' + y = 0 So, for a = 0 (0 + 1) y' - (2-0) (-1) +8=0 MO!- 10 da da dn So, yo) = -10 Differentiate the DE w.rit. ex to get d ((1+1) y") - d (12-») y') + d ly)=0 >>[(+2) y" + y"] - [(2-1)y" - Y'] + y' = 0 [ using prodent rule of differentiation > (1+i) y"+y" + (1-2)y" + y' + y = 0 OF (+1) "' +(4-1) y' + 2y' = 0 At =0(0+1) yn + 10-1) (-10) + 2(-1) = 0 - y" (o)= -8

8 10 So, y = 8 – 1-2 2 C 6

$Or, y = 8 -x -5x^2 - \frac{4}{3}x^3$

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Your answer is exactly correct. Only mistake is that you have used 5 non-zero terms whereas question has asked you to enter only 4 non-zero terms. The first term y(0) = 8 is a constant but you still need to count it as a non-zero term.