ODE
\[ x \left (\text {a0}+x^2\right ) y''(x)+\left (\text {a1}+\text {b1} x^2\right ) y'(x)+\text {a2} x y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.486691 (sec), leaf count = 177
\[\left \{\left \{y(x)\to c_2 \text {a0}^{\frac {1}{2} \left (\frac {\text {a1}}{\text {a0}}-1\right )} x^{1-\frac {\text {a1}}{\text {a0}}} \, _2F_1\left (\frac {\text {a0} \left (\text {b1}-\sqrt {(\text {b1}-1)^2-4 \text {a2}}+1\right )-2 \text {a1}}{4 \text {a0}},\frac {\text {a0} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {a2}}+1\right )-2 \text {a1}}{4 \text {a0}};\frac {3}{2}-\frac {\text {a1}}{2 \text {a0}};-\frac {x^2}{\text {a0}}\right )+c_1 \, _2F_1\left (\frac {1}{4} \left (\text {b1}-\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right ),\frac {1}{4} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right );\frac {\text {a0}+\text {a1}}{2 \text {a0}};-\frac {x^2}{\text {a0}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.192 (sec), leaf count = 184
\[ \left \{ y \left ( x \right ) = \left ( {x}^{2}+{\it a0} \right ) ^{{\frac { \left ( -{\it b1}+2 \right ) {\it a0}+{\it a1}}{2\,{\it a0}}}} \left ( {\mbox {$_2$F$_1$}(-{\frac {1}{4}\sqrt {{{\it b1}}^{2}-4\,{\it a2}-2\,{\it b1}+1}}-{\frac {{\it b1}}{4}}+{\frac {5}{4}},{\frac {1}{4}\sqrt {{{\it b1}}^{2}-4\,{\it a2}-2\,{\it b1}+1}}-{\frac {{\it b1}}{4}}+{\frac {5}{4}};\,{\frac {3\,{\it a0}-{\it a1}}{2\,{\it a0}}};\,-{\frac {{x}^{2}}{{\it a0}}})}{x}^{{\frac {{\it a0}-{\it a1}}{{\it a0}}}}{\it \_C1}+{\mbox {$_2$F$_1$}(-{\frac {1}{4\,{\it a0}} \left ( \sqrt {{{\it b1}}^{2}-4\,{\it a2}-2\,{\it b1}+1}{\it a0}+{\it b1}\,{\it a0}-3\,{\it a0}-2\,{\it a1} \right ) },{\frac {1}{4\,{\it a0}} \left ( \sqrt {{{\it b1}}^{2}-4\,{\it a2}-2\,{\it b1}+1}{\it a0}+ \left ( -{\it b1}+3 \right ) {\it a0}+2\,{\it a1} \right ) };\,{\frac {{\it a0}+{\it a1}}{2\,{\it a0}}};\,-{\frac {{x}^{2}}{{\it a0}}})}{\it \_C2} \right ) \right \} \] Mathematica raw input
DSolve[a2*x*y[x] + (a1 + b1*x^2)*y'[x] + x*(a0 + x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Hypergeometric2F1[(-1 - Sqrt[-4*a2 + (-1 + b1)^2] + b1)/4, (-1 + Sqrt[-4*a2 + (-1 + b1)^2] + b1)/4, (a0 + a1)/(2*a0), -(x^2/a0)] + a0^((-1 + a1/a0)/2)*x^(1 - a1/a0)*C[2]*Hypergeometric2F1[(-2*a1 + a0*(1 - Sqrt[-4*a2 + (-1 + b1)^2] + b1))/(4*a0), (-2*a1 + a0*(1 + Sqrt[-4*a2 + (-1 + b1)^2] + b1))/(4*a0), 3/2 - a1/(2*a0), -(x^2/a0)]}}
Maple raw input
dsolve(x*(x^2+a0)*diff(diff(y(x),x),x)+(b1*x^2+a1)*diff(y(x),x)+a2*x*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (x^2+a0)^(1/2*((-b1+2)*a0+a1)/a0)*(hypergeom([-1/4*(b1^2-4*a2-2*b1+1)^(1/2)-1/4*b1+5/4, 1/4*(b1^2-4*a2-2*b1+1)^(1/2)-1/4*b1+5/4],[1/2*(3*a0-a1)/a0],-1/a0*x^2)*x^((a0-a1)/a0)*_C1+hypergeom([-1/4/a0*((b1^2-4*a2-2*b1+1)^(1/2)*a0+b1*a0-3*a0-2*a1), 1/4*((b1^2-4*a2-2*b1+1)^(1/2)*a0+(-b1+3)*a0+2*a1)/a0],[1/2/a0*(a0+a1)],-1/a0*x^2)*_C2)