Tom N. answered • 06/07/19
Strong proficiency in elementary and advanced mathematics
First divide through by 8x2. let P(x)= 5/4x and Q(x)= (x-1)/8 so function is analytic. let y= xλ∑0∞anxn taking the derivative of this series for y' and y" and then substituting the expansions back into the original equation and factoring out xλ and simplifying gives [8λ2 +2λ -1]a0 + x[(8λ2 +18λ + 9)a1 +a0] + .....xn{ [8(λ +n)2 + 2(λ+n) -1]an + an-1} +....= 0 now equate all factors to 0 to get (8λ2 +2λ -1) =0 and for n≥1 [ 4(λ+n)-1][2(λ +n) +1]an +an-1 =0.
Solve for an .
Factor 8λ2 + 2λ - 1=0 this gives λ1=1/4 and λ2 = -1/2 For λ=1/4 an = - an-1 / 2n( 4n+3). So a1 = -a0/14
a2 =-a1/44 = a0/166 λ = -1/2 an= -an-1/2n(4n-3) a1=-a0/2 a2 =-a1/20 = a0/40. Using these terms in y1(x) and y2(x) the solution is y(x) = cy1(x) + dy2(x) = c x1/4( 1- x/14 +x2/166 +...) +dx-1/2( 1- x/2 + x2/40 +...)
