ODE
\[ a x y'(x)+x^2 y(x) \left (\text {a1}+\text {b1} x^2\right )+x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0917486 (sec), leaf count = 136
\[\left \{\left \{y(x)\to 2^{\frac {a+1}{4}} x^{\frac {1}{2} (-a-1)} \left (x^2\right )^{\frac {a+1}{4}} e^{\frac {1}{2} i \sqrt {\text {b1}} x^2} \left (c_1 U\left (\frac {1}{4} \left (a-\frac {i \text {a1}}{\sqrt {\text {b1}}}+1\right ),\frac {a+1}{2},-i \sqrt {\text {b1}} x^2\right )+c_2 L_{\frac {1}{4} \left (-a+\frac {i \text {a1}}{\sqrt {\text {b1}}}-1\right )}^{\frac {a-1}{2}}\left (-i \sqrt {\text {b1}} x^2\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.185 (sec), leaf count = 86
\[ \left \{ y \left ( x \right ) ={{\rm e}^{-{\frac {i}{2}}\sqrt {{\it b1}}{x}^{2}}} \left ( {{\sl U}\left ({\frac {1}{4} \left ( a\sqrt {{\it b1}}+i{\it a1}+\sqrt {{\it b1}} \right ) {\frac {1}{\sqrt {{\it b1}}}}},\,{\frac {1}{2}}+{\frac {a}{2}},\,i\sqrt {{\it b1}}{x}^{2}\right )}{\it \_C2}+{{\sl M}\left ({\frac {1}{4} \left ( a\sqrt {{\it b1}}+i{\it a1}+\sqrt {{\it b1}} \right ) {\frac {1}{\sqrt {{\it b1}}}}},\,{\frac {1}{2}}+{\frac {a}{2}},\,i\sqrt {{\it b1}}{x}^{2}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[x^2*(a1 + b1*x^2)*y[x] + a*x*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> 2^((1 + a)/4)*E^((I/2)*Sqrt[b1]*x^2)*x^((-1 - a)/2)*(x^2)^((1 + a)/4)*(C[1]*HypergeometricU[(1 + a - (I*a1)/Sqrt[b1])/4, (1 + a)/2, (-I)*Sqrt[b1]*x^2] + C[2]*LaguerreL[(-1 - a + (I*a1)/Sqrt[b1])/4, (-1 + a)/2, (-I)*Sqrt[b1]*x^2])}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+x^2*(b1*x^2+a1)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = exp(-1/2*I*b1^(1/2)*x^2)*(KummerU(1/4*(a*b1^(1/2)+I*a1+b1^(1/2))/b1^(1/2),1/2+1/2*a,I*b1^(1/2)*x^2)*_C2+KummerM(1/4*(a*b1^(1/2)+I*a1+b1^(1/2))/b1^(1/2),1/2+1/2*a,I*b1^(1/2)*x^2)*_C1)