4.30.2 \(-y(x) \left (a+b x+c x^2\right )+x^2 y''(x)+x y'(x)=0\)

ODE
\[ -y(x) \left (a+b x+c x^2\right )+x^2 y''(x)+x y'(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.182095 (sec), leaf count = 104

\[\left \{\left \{y(x)\to x^{\sqrt {a}} e^{-\sqrt {c} x} \left (c_1 U\left (\frac {b}{2 \sqrt {c}}+\sqrt {a}+\frac {1}{2},2 \sqrt {a}+1,2 \sqrt {c} x\right )+c_2 L_{-\frac {b}{2 \sqrt {c}}-\sqrt {a}-\frac {1}{2}}^{2 \sqrt {a}}\left (2 \sqrt {c} x\right )\right )\right \}\right \}\]

Maple ✓
cpu = 0.847 (sec), leaf count = 47

\[\left [y \left (x \right ) = \frac {\textit {\_C1} \WhittakerM \left (-\frac {b}{2 \sqrt {c}}, \sqrt {a}, 2 \sqrt {c}\, x \right )}{\sqrt {x}}+\frac {\textit {\_C2} \WhittakerW \left (-\frac {b}{2 \sqrt {c}}, \sqrt {a}, 2 \sqrt {c}\, x \right )}{\sqrt {x}}\right ]\] Mathematica raw input

DSolve[-((a + b*x + c*x^2)*y[x]) + x*y'[x] + x^2*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (x^Sqrt[a]*(C[1]*HypergeometricU[1/2 + Sqrt[a] + b/(2*Sqrt[c]), 1 + 2*
Sqrt[a], 2*Sqrt[c]*x] + C[2]*LaguerreL[-1/2 - Sqrt[a] - b/(2*Sqrt[c]), 2*Sqrt[a]
, 2*Sqrt[c]*x]))/E^(Sqrt[c]*x)}}

Maple raw input

dsolve(x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)-(c*x^2+b*x+a)*y(x) = 0, y(x))

Maple raw output

[y(x) = _C1/x^(1/2)*WhittakerM(-1/2*b/c^(1/2),a^(1/2),2*c^(1/2)*x)+_C2/x^(1/2)*W
hittakerW(-1/2*b/c^(1/2),a^(1/2),2*c^(1/2)*x)]