4.34.12 \(\text {a1} x y'(x)+y(x) (\text {a2}+\text {b2} x)+x^3 y''(x)=0\)

ODE
\[ \text {a1} x y'(x)+y(x) (\text {a2}+\text {b2} x)+x^3 y''(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0946935 (sec), leaf count = 190

\[\left \{\left \{y(x)\to -i i^{-\sqrt {1-4 \text {b2}}} \text {a1}^{-\frac {1}{2} \sqrt {1-4 \text {b2}}-\frac {1}{2}} \left (\frac {1}{x}\right )^{-\frac {1}{2} \sqrt {1-4 \text {b2}}-\frac {1}{2}} \left (c_1 \, _1F_1\left (-\frac {\sqrt {1-4 \text {b2}} \text {a1}+\text {a1}+2 \text {a2}}{2 \text {a1}};1-\sqrt {1-4 \text {b2}};\frac {\text {a1}}{x}\right )+i^{2 \sqrt {1-4 \text {b2}}} c_2 \text {a1}^{\sqrt {1-4 \text {b2}}} \left (\frac {1}{x}\right )^{\sqrt {1-4 \text {b2}}} \, _1F_1\left (\frac {1}{2} \left (-\frac {2 \text {a2}}{\text {a1}}+\sqrt {1-4 \text {b2}}-1\right );\sqrt {1-4 \text {b2}}+1;\frac {\text {a1}}{x}\right )\right )\right \}\right \}\]

Maple ✓
cpu = 0.219 (sec), leaf count = 96

\[ \left \{ y \left ( x \right ) ={x}^{-{\frac {1}{2}\sqrt {1-4\,{\it b2}}}-{\frac {1}{2}}}x \left ( {{\sl U}\left ({\frac {1}{2\,{\it a1}} \left ( \sqrt {1-4\,{\it b2}}{\it a1}-{\it a1}-2\,{\it a2} \right ) },\,1+\sqrt {1-4\,{\it b2}},\,{\frac {{\it a1}}{x}}\right )}{\it \_C2}+{{\sl M}\left ({\frac {1}{2\,{\it a1}} \left ( \sqrt {1-4\,{\it b2}}{\it a1}-{\it a1}-2\,{\it a2} \right ) },\,1+\sqrt {1-4\,{\it b2}},\,{\frac {{\it a1}}{x}}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input

DSolve[(a2 + b2*x)*y[x] + a1*x*y'[x] + x^3*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> ((-I)*a1^(-1/2 - Sqrt[1 - 4*b2]/2)*(x^(-1))^(-1/2 - Sqrt[1 - 4*b2]/2)*
(I^(2*Sqrt[1 - 4*b2])*a1^Sqrt[1 - 4*b2]*(x^(-1))^Sqrt[1 - 4*b2]*C[2]*Hypergeomet
ric1F1[(-1 - (2*a2)/a1 + Sqrt[1 - 4*b2])/2, 1 + Sqrt[1 - 4*b2], a1/x] + C[1]*Hyp
ergeometric1F1[-(a1 + 2*a2 + a1*Sqrt[1 - 4*b2])/(2*a1), 1 - Sqrt[1 - 4*b2], a1/x
]))/I^Sqrt[1 - 4*b2]}}

Maple raw input

dsolve(x^3*diff(diff(y(x),x),x)+a1*x*diff(y(x),x)+(b2*x+a2)*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = x^(-1/2*(1-4*b2)^(1/2)-1/2)*x*(KummerU(1/2*((1-4*b2)^(1/2)*a1-a1-2*a2)/a1
,1+(1-4*b2)^(1/2),1/x*a1)*_C2+KummerM(1/2*((1-4*b2)^(1/2)*a1-a1-2*a2)/a1,1+(1-4*
b2)^(1/2),1/x*a1)*_C1)