ODE
\[ x^2 (\text {a0}+x) y''(x)+x (\text {a1}+\text {b1} x) y'(x)+y(x) (\text {a2}+\text {b2} x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.577661 (sec), leaf count = 391
\[\left \{\left \{y(x)\to \text {a0}^{-\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}+\text {a0}-\text {a1}}{2 \text {a0}}} x^{-\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}-\text {a0}+\text {a1}}{2 \text {a0}}} \left (c_2 x^{\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}}{\text {a0}}} \, _2F_1\left (\frac {-\text {a1}+\text {a0} \left (\text {b1}+\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}\right )+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{2 \text {a0}},\frac {\text {b1} \text {a0}-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1} \text {a0}-\text {a1}+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{2 \text {a0}};\frac {\text {a0}+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{\text {a0}};-\frac {x}{\text {a0}}\right )+c_1 \text {a0}^{\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}}{\text {a0}}} \, _2F_1\left (-\frac {\text {a1}-\text {a0} \left (\text {b1}+\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}\right )+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{2 \text {a0}},-\frac {-\text {b1} \text {a0}+\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1} \text {a0}+\text {a1}+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{2 \text {a0}};1-\frac {\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{\text {a0}};-\frac {x}{\text {a0}}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.251 (sec), leaf count = 347
\[ \left \{ y \left ( x \right ) = \left ( {\it a0}+x \right ) ^{{\frac {-{\it b1}\,{\it a0}+{\it a0}+{\it a1}}{{\it a0}}}} \left ( {x}^{{\frac {1}{2\,{\it a0}} \left ( -{\it a1}+{\it a0}+\sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}} \right ) }}{\mbox {$_2$F$_1$}({\frac {1}{2\,{\it a0}} \left ( \sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}}+\sqrt {{{\it b1}}^{2}-2\,{\it b1}-4\,{\it b2}+1}{\it a0}+ \left ( -{\it b1}+2 \right ) {\it a0}+{\it a1} \right ) },{\frac {1}{2\,{\it a0}} \left ( \sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}}-\sqrt {{{\it b1}}^{2}-2\,{\it b1}-4\,{\it b2}+1}{\it a0}+ \left ( -{\it b1}+2 \right ) {\it a0}+{\it a1} \right ) };\,{\frac {1}{{\it a0}} \left ( {\it a0}+\sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}} \right ) };\,-{\frac {x}{{\it a0}}})}{\it \_C1}+{x}^{{\frac {1}{2\,{\it a0}} \left ( -{\it a1}+{\it a0}-\sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}} \right ) }}{\mbox {$_2$F$_1$}(-{\frac {1}{2\,{\it a0}} \left ( \sqrt {{{\it b1}}^{2}-2\,{\it b1}-4\,{\it b2}+1}{\it a0}+{\it b1}\,{\it a0}+\sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}}-2\,{\it a0}-{\it a1} \right ) },{\frac {1}{2\,{\it a0}} \left ( -\sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}}+\sqrt {{{\it b1}}^{2}-2\,{\it b1}-4\,{\it b2}+1}{\it a0}+ \left ( -{\it b1}+2 \right ) {\it a0}+{\it a1} \right ) };\,{\frac {1}{{\it a0}} \left ( {\it a0}-\sqrt {{{\it a0}}^{2}+ \left ( -2\,{\it a1}-4\,{\it a2} \right ) {\it a0}+{{\it a1}}^{2}} \right ) };\,-{\frac {x}{{\it a0}}})}{\it \_C2} \right ) \right \} \] 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*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), 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 + Sqrt[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)])/(a0^((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)*a0+a1)/a0],[(a0-(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2))/a0],-1/a0*x)*_C2)