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 = 0.320568 (sec), leaf count = 190
\[\left \{\left \{y(x)\to -i i^{-\sqrt {1-4 \text {b2}}} \text {a1}^{-\frac {1}{2} \sqrt {1-4 \text {b2}}-\frac {1}{2}} \left (\frac {1}{x}\right )^{-\frac {1}{2} \sqrt {1-4 \text {b2}}-\frac {1}{2}} \left (i^{2 \sqrt {1-4 \text {b2}}} c_2 \text {a1}^{\sqrt {1-4 \text {b2}}} \left (\frac {1}{x}\right )^{\sqrt {1-4 \text {b2}}} \, _1F_1\left (\frac {1}{2} \left (-\frac {2 \text {a2}}{\text {a1}}+\sqrt {1-4 \text {b2}}-1\right );\sqrt {1-4 \text {b2}}+1;\frac {\text {a1}}{x}\right )+c_1 \, _1F_1\left (-\frac {\sqrt {1-4 \text {b2}} \text {a1}+\text {a1}+2 \text {a2}}{2 \text {a1}};1-\sqrt {1-4 \text {b2}};\frac {\text {a1}}{x}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.549 (sec), leaf count = 109
\[\left [y \left (x \right ) = \textit {\_C1} \,x^{-\frac {\sqrt {1-4 \mathit {b2}}}{2}-\frac {1}{2}} x \KummerM \left (\frac {\sqrt {1-4 \mathit {b2}}\, \mathit {a1} -\mathit {a1} -2 \mathit {a2}}{2 \mathit {a1}}, 1+\sqrt {1-4 \mathit {b2}}, \frac {\mathit {a1}}{x}\right )+\textit {\_C2} \,x^{-\frac {\sqrt {1-4 \mathit {b2}}}{2}-\frac {1}{2}} x \KummerU \left (\frac {\sqrt {1-4 \mathit {b2}}\, \mathit {a1} -\mathit {a1} -2 \mathit {a2}}{2 \mathit {a1}}, 1+\sqrt {1-4 \mathit {b2}}, \frac {\mathit {a1}}{x}\right )\right ]\] 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]*Hypergeometric1F1[(-1 - (2*a2)/a1 + Sqrt[1 - 4*b2])/2, 1 + Sqrt[1 - 4*b2], a1/x] + C[1]*Hypergeometric1F1[-1/2*(a1 + 2*a2 + a1*Sqrt[1 - 4*b2])/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))
Maple raw output
[y(x) = _C1*x^(-1/2*(1-4*b2)^(1/2)-1/2)*x*KummerM(1/2*((1-4*b2)^(1/2)*a1-a1-2*a2)/a1,1+(1-4*b2)^(1/2),1/x*a1)+_C2*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)]