4.34.43 x2(a0+x)y(x)+x(a1+b1x)y(x)+y(x)(a2+b2x)=0

ODE
x2(a0+x)y(x)+x(a1+b1x)y(x)+y(x)(a2+b2x)=0 ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 0.577661 (sec), leaf count = 391

{{y(x)a0a022a0(a1+2a2)+a12+a0a12a0xa022a0(a1+2a2)+a12a0+a12a0(c2xa022a0(a1+2a2)+a12a02F1(a1+a0(b1+b122b14b2+1)+a022(a1+2a2)a0+a122a0,b1a0b122b14b2+1a0a1+a022(a1+2a2)a0+a122a0;a0+a022(a1+2a2)a0+a12a0;xa0)+c1a0a022a0(a1+2a2)+a12a02F1(a1a0(b1+b122b14b2+1)+a022(a1+2a2)a0+a122a0,b1a0+b122b14b2+1a0+a1+a022(a1+2a2)a0+a122a0;1a022(a1+2a2)a0+a12a0;xa0))}}

Maple
cpu = 0.251 (sec), leaf count = 347

{y(x)=(a0+x)b1a0+a0+a1a0(x12a0(a1+a0+a02+(2a14a2)a0+a12)2F1(12a0(a02+(2a14a2)a0+a12+b122b14b2+1a0+(b1+2)a0+a1),12a0(a02+(2a14a2)a0+a12b122b14b2+1a0+(b1+2)a0+a1);1a0(a0+a02+(2a14a2)a0+a12);xa0)_C1+x12a0(a1+a0a02+(2a14a2)a0+a12)2F1(12a0(b122b14b2+1a0+b1a0+a02+(2a14a2)a0+a122a0a1),12a0(a02+(2a14a2)a0+a12+b122b14b2+1a0+(b1+2)a0+a1);1a0(a0a02+(2a14a2)a0+a12);xa0)_C2)} Mathematica raw input

DSolve[(a2 + b2*x)*y[x] + x*(a1 + b1*x)*y'[x] + x^2*(a0 + x)*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (a0^(Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)]/a0)*C[1]*Hypergeometric2F1[-
(a1 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)] - a0*(b1 + Sqrt[1 - 2*b1 + b1^2 - 4*b
2]))/(2*a0), -(a1 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)] - a0*b1 + a0*Sqrt[1 - 2
*b1 + b1^2 - 4*b2])/(2*a0), 1 - Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)]/a0, -(x/a0)
] + x^(Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)]/a0)*C[2]*Hypergeometric2F1[(-a1 + Sq
rt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)] + a0*(b1 + Sqrt[1 - 2*b1 + b1^2 - 4*b2]))/(2*
a0), (-a1 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)] + a0*b1 - a0*Sqrt[1 - 2*b1 + b1
^2 - 4*b2])/(2*a0), (a0 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)])/a0, -(x/a0)])/(a
0^((a0 - a1 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)])/(2*a0))*x^((-a0 + a1 + Sqrt[
a0^2 + a1^2 - 2*a0*(a1 + 2*a2)])/(2*a0)))}}

Maple raw input

dsolve(x^2*(a0+x)*diff(diff(y(x),x),x)+x*(b1*x+a1)*diff(y(x),x)+(b2*x+a2)*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = (a0+x)^((-a0*b1+a0+a1)/a0)*(x^(1/2*(-a1+a0+(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1
/2))/a0)*hypergeom([1/2*((a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2)+(b1^2-2*b1-4*b2+1)^(1
/2)*a0+(-b1+2)*a0+a1)/a0, 1/2*((a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2)-(b1^2-2*b1-4*b2
+1)^(1/2)*a0+(-b1+2)*a0+a1)/a0],[1/a0*(a0+(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2))],-1
/a0*x)*_C1+x^(1/2*(-a1+a0-(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2))/a0)*hypergeom([-1/2
/a0*((b1^2-2*b1-4*b2+1)^(1/2)*a0+b1*a0+(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2)-2*a0-a1
), 1/2*(-(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2)+(b1^2-2*b1-4*b2+1)^(1/2)*a0+(-b1+2)*a
0+a1)/a0],[(a0-(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2))/a0],-1/a0*x)*_C2)