ODE
\[ \text {a1} x y'(x)+y(x) \left (\text {a2}+\text {b2} x+\text {c2} x^2\right )+x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.215666 (sec), leaf count = 169
\[\left \{\left \{y(x)\to e^{-i \sqrt {\text {c2}} x} x^{\frac {1}{2} \left (\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}-\text {a1}+1\right )} \left (c_1 U\left (\frac {1}{2} \left (\frac {i \text {b2}}{\sqrt {\text {c2}}}+\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}+1\right ),\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}+1,2 i \sqrt {\text {c2}} x\right )+c_2 L_{\frac {1}{2} \left (-\frac {i \text {b2}}{\sqrt {\text {c2}}}-\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}-1\right )}^{\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}}\left (2 i \sqrt {\text {c2}} x\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.492 (sec), leaf count = 79
\[\left [y \left (x \right ) = \textit {\_C1} \,x^{-\frac {\mathit {a1}}{2}} \WhittakerM \left (-\frac {i \mathit {b2}}{2 \sqrt {\mathit {c2}}}, \frac {\sqrt {\mathit {a1}^{2}-2 \mathit {a1} -4 \mathit {a2} +1}}{2}, 2 i \sqrt {\mathit {c2}}\, x \right )+\textit {\_C2} \,x^{-\frac {\mathit {a1}}{2}} \WhittakerW \left (-\frac {i \mathit {b2}}{2 \sqrt {\mathit {c2}}}, \frac {\sqrt {\mathit {a1}^{2}-2 \mathit {a1} -4 \mathit {a2} +1}}{2}, 2 i \sqrt {\mathit {c2}}\, x \right )\right ]\] Mathematica raw input
DSolve[(a2 + b2*x + c2*x^2)*y[x] + a1*x*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (x^((1 - a1 + Sqrt[1 - 2*a1 + a1^2 - 4*a2])/2)*(C[1]*HypergeometricU[(1 + Sqrt[1 - 2*a1 + a1^2 - 4*a2] + (I*b2)/Sqrt[c2])/2, 1 + Sqrt[1 - 2*a1 + a1^2 - 4*a2], (2*I)*Sqrt[c2]*x] + C[2]*LaguerreL[(-1 - Sqrt[1 - 2*a1 + a1^2 - 4*a2] - (I*b2)/Sqrt[c2])/2, Sqrt[1 - 2*a1 + a1^2 - 4*a2], (2*I)*Sqrt[c2]*x]))/E^(I*Sqrt[c2]*x)}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+a1*x*diff(y(x),x)+(c2*x^2+b2*x+a2)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*x^(-1/2*a1)*WhittakerM(-1/2*I*b2/c2^(1/2),1/2*(a1^2-2*a1-4*a2+1)^(1/2),2*I*c2^(1/2)*x)+_C2*x^(-1/2*a1)*WhittakerW(-1/2*I*b2/c2^(1/2),1/2*(a1^2-2*a1-4*a2+1)^(1/2),2*I*c2^(1/2)*x)]