ODE
\[ \left (\text {a1}+\text {b1} x^2\right ) y'(x)+\text {a2} x y(x)+x^3 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.102556 (sec), leaf count = 283
\[\left \{\left \{y(x)\to (-1)^{\frac {1}{4} \left (-\sqrt {(\text {b1}-1)^2-4 \text {a2}}+\text {b1}+7\right )} 2^{\frac {1}{4} \left (-\sqrt {(\text {b1}-1)^2-4 \text {a2}}-\text {b1}+1\right )} \text {a1}^{\frac {1}{4} \left (-\sqrt {(\text {b1}-1)^2-4 \text {a2}}+\text {b1}-1\right )} \left (\frac {1}{x}\right )^{\frac {1}{2} \left (-\sqrt {(\text {b1}-1)^2-4 \text {a2}}+\text {b1}-1\right )} \left (c_1 2^{\frac {1}{2} \sqrt {(\text {b1}-1)^2-4 \text {a2}}} \, _1F_1\left (\frac {1}{4} \left (\text {b1}-\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right );1-\frac {1}{2} \sqrt {(\text {b1}-1)^2-4 \text {a2}};\frac {\text {a1}}{2 x^2}\right )+c_2 i^{\sqrt {(\text {b1}-1)^2-4 \text {a2}}} \text {a1}^{\frac {1}{2} \sqrt {(\text {b1}-1)^2-4 \text {a2}}} \left (\frac {1}{x}\right )^{\sqrt {(\text {b1}-1)^2-4 \text {a2}}} \, _1F_1\left (\frac {1}{4} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right );\frac {1}{2} \left (\sqrt {(\text {b1}-1)^2-4 \text {a2}}+2\right );\frac {\text {a1}}{2 x^2}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.255 (sec), leaf count = 120
\[ \left \{ y \left ( x \right ) ={x}^{-{\frac {{\it b1}}{2}}+{\frac {1}{2}}-{\frac {1}{2}\sqrt {{{\it b1}}^{2}-4\,{\it a2}-2\,{\it b1}+1}}} \left ( {{\sl U}\left (-{\frac {1}{4}}+{\frac {1}{4}\sqrt {{{\it b1}}^{2}-4\,{\it a2}-2\,{\it b1}+1}}+{\frac {{\it b1}}{4}},\,1+{\frac {1}{2}\sqrt {{{\it b1}}^{2}-4\,{\it a2}-2\,{\it b1}+1}},\,{\frac {{\it a1}}{2\,{x}^{2}}}\right )}{\it \_C2}+{{\sl M}\left (-{\frac {1}{4}}+{\frac {1}{4}\sqrt {{{\it b1}}^{2}-4\,{\it a2}-2\,{\it b1}+1}}+{\frac {{\it b1}}{4}},\,1+{\frac {1}{2}\sqrt {{{\it b1}}^{2}-4\,{\it a2}-2\,{\it b1}+1}},\,{\frac {{\it a1}}{2\,{x}^{2}}}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[a2*x*y[x] + (a1 + b1*x^2)*y'[x] + x^3*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-1)^((7 - Sqrt[-4*a2 + (-1 + b1)^2] + b1)/4)*2^((1 - Sqrt[-4*a2 + (-1 + b1)^2] - b1)/4)*a1^((-1 - Sqrt[-4*a2 + (-1 + b1)^2] + b1)/4)*(x^(-1))^((-1 - Sqrt[-4*a2 + (-1 + b1)^2] + b1)/2)*(2^(Sqrt[-4*a2 + (-1 + b1)^2]/2)*C[1]*Hypergeometric1F1[(-1 - Sqrt[-4*a2 + (-1 + b1)^2] + b1)/4, 1 - Sqrt[-4*a2 + (-1 + b1)^2]/2, a1/(2*x^2)] + I^Sqrt[-4*a2 + (-1 + b1)^2]*a1^(Sqrt[-4*a2 + (-1 + b1)^2]/2)*(x^(-1))^Sqrt[-4*a2 + (-1 + b1)^2]*C[2]*Hypergeometric1F1[(-1 + Sqrt[-4*a2 + (-1 + b1)^2] + b1)/4, (2 + Sqrt[-4*a2 + (-1 + b1)^2])/2, a1/(2*x^2)])}}
Maple raw input
dsolve(x^3*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^(-1/2*b1+1/2-1/2*(b1^2-4*a2-2*b1+1)^(1/2))*(KummerU(-1/4+1/4*(b1^2-4*a2-2*b1+1)^(1/2)+1/4*b1,1+1/2*(b1^2-4*a2-2*b1+1)^(1/2),1/2*a1/x^2)*_C2+KummerM(-1/4+1/4*(b1^2-4*a2-2*b1+1)^(1/2)+1/4*b1,1+1/2*(b1^2-4*a2-2*b1+1)^(1/2),1/2*a1/x^2)*_C1)