4.31.8 \(x (\text {a1}+\text {b1} x) y'(x)+y(x) \left (\text {a2}+\text {b2} x+\text {c2} x^2\right )+x^2 y''(x)=0\)

ODE
\[ x (\text {a1}+\text {b1} x) y'(x)+y(x) \left (\text {a2}+\text {b2} x+\text {c2} x^2\right )+x^2 y''(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.244875 (sec), leaf count = 223

\[\left \{\left \{y(x)\to x^{\frac {1}{2} \left (\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}-\text {a1}+1\right )} e^{-\frac {1}{2} x \left (\sqrt {\text {b1}^2-4 \text {c2}}+\text {b1}\right )} \left (c_1 U\left (\frac {\sqrt {\text {b1}^2-4 \text {c2}} \left (\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}+1\right )+\text {a1} \text {b1}-2 \text {b2}}{2 \sqrt {\text {b1}^2-4 \text {c2}}},\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}+1,\sqrt {\text {b1}^2-4 \text {c2}} x\right )+c_2 L_{\frac {-\sqrt {\text {b1}^2-4 \text {c2}} \left (\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}+1\right )-\text {a1} \text {b1}+2 \text {b2}}{2 \sqrt {\text {b1}^2-4 \text {c2}}}}^{\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}}\left (\sqrt {\text {b1}^2-4 \text {c2}} x\right )\right )\right \}\right \}\]

Maple ✓
cpu = 0.65 (sec), leaf count = 119

\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-\frac {\mathit {b1} x}{2}} x^{-\frac {\mathit {a1}}{2}} \WhittakerM \left (-\frac {\mathit {a1} \mathit {b1} -2 \mathit {b2}}{2 \sqrt {\mathit {b1}^{2}-4 \mathit {c2}}}, \frac {\sqrt {\mathit {a1}^{2}-2 \mathit {a1} -4 \mathit {a2} +1}}{2}, \sqrt {\mathit {b1}^{2}-4 \mathit {c2}}\, x \right )+\textit {\_C2} \,{\mathrm e}^{-\frac {\mathit {b1} x}{2}} x^{-\frac {\mathit {a1}}{2}} \WhittakerW \left (-\frac {\mathit {a1} \mathit {b1} -2 \mathit {b2}}{2 \sqrt {\mathit {b1}^{2}-4 \mathit {c2}}}, \frac {\sqrt {\mathit {a1}^{2}-2 \mathit {a1} -4 \mathit {a2} +1}}{2}, \sqrt {\mathit {b1}^{2}-4 \mathit {c2}}\, x \right )\right ]\] Mathematica raw input

DSolve[(a2 + b2*x + c2*x^2)*y[x] + x*(a1 + b1*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[(
a1*b1 - 2*b2 + (1 + Sqrt[1 - 2*a1 + a1^2 - 4*a2])*Sqrt[b1^2 - 4*c2])/(2*Sqrt[b1^
2 - 4*c2]), 1 + Sqrt[1 - 2*a1 + a1^2 - 4*a2], Sqrt[b1^2 - 4*c2]*x] + C[2]*Laguer
reL[(-(a1*b1) + 2*b2 - (1 + Sqrt[1 - 2*a1 + a1^2 - 4*a2])*Sqrt[b1^2 - 4*c2])/(2*
Sqrt[b1^2 - 4*c2]), Sqrt[1 - 2*a1 + a1^2 - 4*a2], Sqrt[b1^2 - 4*c2]*x]))/E^(((b1
 + Sqrt[b1^2 - 4*c2])*x)/2)}}

Maple raw input

dsolve(x^2*diff(diff(y(x),x),x)+x*(b1*x+a1)*diff(y(x),x)+(c2*x^2+b2*x+a2)*y(x) = 0, y(x))

Maple raw output

[y(x) = _C1*exp(-1/2*b1*x)*x^(-1/2*a1)*WhittakerM(-1/2*(a1*b1-2*b2)/(b1^2-4*c2)^
(1/2),1/2*(a1^2-2*a1-4*a2+1)^(1/2),(b1^2-4*c2)^(1/2)*x)+_C2*exp(-1/2*b1*x)*x^(-1
/2*a1)*WhittakerW(-1/2*(a1*b1-2*b2)/(b1^2-4*c2)^(1/2),1/2*(a1^2-2*a1-4*a2+1)^(1/
2),(b1^2-4*c2)^(1/2)*x)]