ODE
\[ a x y'(x)+y(x) \left (b+c x^3\right )+x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.236229 (sec), leaf count = 156
\[\left \{\left \{y(x)\to 3^{\frac {a-1}{3}} c^{\frac {1}{6}-\frac {a}{6}} x^{\frac {1}{2}-\frac {a}{2}} \left (c_1 \Gamma \left (1-\frac {1}{3} \sqrt {a^2-2 a-4 b+1}\right ) J_{-\frac {1}{3} \sqrt {a^2-2 a-4 b+1}}\left (\frac {2}{3} \sqrt {c} x^{3/2}\right )+c_2 \Gamma \left (\frac {1}{3} \sqrt {a^2-2 a-4 b+1}+1\right ) J_{\frac {1}{3} \sqrt {a^2-2 a-4 b+1}}\left (\frac {2}{3} \sqrt {c} x^{3/2}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.044 (sec), leaf count = 71
\[\left [y \left (x \right ) = \textit {\_C1} \,x^{-\frac {a}{2}+\frac {1}{2}} \BesselJ \left (\frac {\sqrt {a^{2}-2 a -4 b +1}}{3}, \frac {2 \sqrt {c}\, x^{\frac {3}{2}}}{3}\right )+\textit {\_C2} \,x^{-\frac {a}{2}+\frac {1}{2}} \BesselY \left (\frac {\sqrt {a^{2}-2 a -4 b +1}}{3}, \frac {2 \sqrt {c}\, x^{\frac {3}{2}}}{3}\right )\right ]\] Mathematica raw input
DSolve[(b + c*x^3)*y[x] + a*x*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> 3^((-1 + a)/3)*c^(1/6 - a/6)*x^(1/2 - a/2)*(BesselJ[-1/3*Sqrt[1 - 2*a + a^2 - 4*b], (2*Sqrt[c]*x^(3/2))/3]*C[1]*Gamma[1 - Sqrt[1 - 2*a + a^2 - 4*b]/3] + BesselJ[Sqrt[1 - 2*a + a^2 - 4*b]/3, (2*Sqrt[c]*x^(3/2))/3]*C[2]*Gamma[1 + Sqrt[1 - 2*a + a^2 - 4*b]/3])}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+(c*x^3+b)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*x^(-1/2*a+1/2)*BesselJ(1/3*(a^2-2*a-4*b+1)^(1/2),2/3*c^(1/2)*x^(3/2))+_C2*x^(-1/2*a+1/2)*BesselY(1/3*(a^2-2*a-4*b+1)^(1/2),2/3*c^(1/2)*x^(3/2))]