ODE
\[ a y'(x)+y(x) \left (b+c x^2\right )+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0262869 (sec), leaf count = 117
\[\left \{\left \{y(x)\to e^{\frac {1}{2} x \left (-a-i \sqrt {c} x\right )} \left (c_1 H_{\frac {i \left (a^2-4 b+4 i \sqrt {c}\right )}{8 \sqrt {c}}}\left (\sqrt [4]{-1} \sqrt [4]{c} x\right )+c_2 \, _1F_1\left (-\frac {i \left (a^2-4 b+4 i \sqrt {c}\right )}{16 \sqrt {c}};\frac {1}{2};i \sqrt {c} x^2\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.227 (sec), leaf count = 88
\[ \left \{ y \left ( x \right ) ={{\rm e}^{-{\frac {x}{2} \left ( i\sqrt {c}x+a \right ) }}}x \left ( {{\sl U}\left (-{\frac {1}{16} \left ( i{a}^{2}-4\,ib-12\,\sqrt {c} \right ) {\frac {1}{\sqrt {c}}}},\,{\frac {3}{2}},\,i\sqrt {c}{x}^{2}\right )}{\it \_C2}+{{\sl M}\left (-{\frac {1}{16} \left ( i{a}^{2}-4\,ib-12\,\sqrt {c} \right ) {\frac {1}{\sqrt {c}}}},\,{\frac {3}{2}},\,i\sqrt {c}{x}^{2}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[(b + c*x^2)*y[x] + a*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> E^((x*(-a - I*Sqrt[c]*x))/2)*(C[1]*HermiteH[((I/8)*(a^2 - 4*b + (4*I)*Sqrt[c]))/Sqrt[c], (-1)^(1/4)*c^(1/4)*x] + C[2]*Hypergeometric1F1[((-I/16)*(a^2 - 4*b + (4*I)*Sqrt[c]))/Sqrt[c], 1/2, I*Sqrt[c]*x^2])}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)+(c*x^2+b)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = exp(-1/2*x*(I*c^(1/2)*x+a))*x*(KummerU(-1/16*(I*a^2-4*I*b-12*c^(1/2))/c^(1/2),3/2,I*c^(1/2)*x^2)*_C2+KummerM(-1/16*(I*a^2-4*I*b-12*c^(1/2))/c^(1/2),3/2,I*c^(1/2)*x^2)*_C1)