##### 4.25.13 $$y(x) \left (a+b x+c x^2\right )+y''(x)=0$$

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

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 0.163496 (sec), leaf count = 110

$\left \{\left \{y(x)\to c_2 D_{\frac {-i b^2-4 c^{3/2}+4 i a c}{8 c^{3/2}}}\left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (b+2 c x)}{c^{3/4}}\right )+c_1 D_{\frac {i \left (b^2+4 i c^{3/2}-4 a c\right )}{8 c^{3/2}}}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (b+2 c x)}{c^{3/4}}\right )\right \}\right \}$

Maple
cpu = 0.283 (sec), leaf count = 121

$\left [y \left (x \right ) = \textit {\_C1} \hypergeom \left (\left [\frac {4 i a c -i b^{2}+4 c^{\frac {3}{2}}}{16 c^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 c x +b \right )^{2}}{4 c^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {i x \left (c x +b \right )}{2 \sqrt {c}}}+\textit {\_C2} \left (2 c x +b \right ) \hypergeom \left (\left [\frac {4 i a c -i b^{2}+12 c^{\frac {3}{2}}}{16 c^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 c x +b \right )^{2}}{4 c^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {i x \left (c x +b \right )}{2 \sqrt {c}}}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> C[2]*ParabolicCylinderD[((-I)*b^2 + (4*I)*a*c - 4*c^(3/2))/(8*c^(3/2))
, ((-1/2 + I/2)*(b + 2*c*x))/c^(3/4)] + C[1]*ParabolicCylinderD[((I/8)*(b^2 - 4*
a*c + (4*I)*c^(3/2)))/c^(3/2), ((1/2 + I/2)*(b + 2*c*x))/c^(3/4)]}}

Maple raw input

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

Maple raw output

[y(x) = _C1*hypergeom([1/16*(4*I*a*c-I*b^2+4*c^(3/2))/c^(3/2)],[1/2],1/4*I*(2*c*
x+b)^2/c^(3/2))*exp(-1/2*I*x*(c*x+b)/c^(1/2))+_C2*(2*c*x+b)*hypergeom([1/16*(4*I
*a*c-I*b^2+12*c^(3/2))/c^(3/2)],[3/2],1/4*I*(2*c*x+b)^2/c^(3/2))*exp(-1/2*I*x*(c
*x+b)/c^(1/2))]