> de:=x*(x-1)*(D@@2)(y)(x)-D(y)(x)+2*x*y(x)=0;

>

de := x*(x-1)*`@@`(D,2)(y)(x)-D(y)(x)+2*x*y(x) = 0

> Order:=8;

Order := 8

> dsolve(de,y(x),series);

y(x) = _C1*(series(1+1/2*x^2+1/9*x^3+5/48*x^4+53/90...
y(x) = _C1*(series(1+1/2*x^2+1/9*x^3+5/48*x^4+53/90...
y(x) = _C1*(series(1+1/2*x^2+1/9*x^3+5/48*x^4+53/90...

> y1:=subs(_C1=1,_C2=0,");

y1 := y(x) = series(1+1/2*x^2+1/9*x^3+5/48*x^4+53/9...

> y11:=convert(rhs(y1),polynom);

y11 := 1+1/2*x^2+1/9*x^3+5/48*x^4+53/900*x^5+499/12...

> ysq:=y11*y11;

ysq := (1+1/2*x^2+1/9*x^3+5/48*x^4+53/900*x^5+499/1...

> num:=simplify(exp(int(1/x/(x-1),x)));

num := 1/x*(x-1)

> in1:=int(convert(taylor((x-1)/ysq,x,8),polynom)/x,x);

in1 := -70087/5556600*x^7+7177/97200*x^6+587/9000*x...

> y2:=simplify(y11*in1);

y2 := -1/1058598979200000*(3175200+1587600*x^2+3528...
y2 := -1/1058598979200000*(3175200+1587600*x^2+3528...
y2 := -1/1058598979200000*(3175200+1587600*x^2+3528...
y2 := -1/1058598979200000*(3175200+1587600*x^2+3528...

> plot({y11,y2},x=.1..1);

[Maple Plot]

>

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