ODE
\[ a x^2 y'(x)+y(x) (b+c x)+x^3 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.195936 (sec), leaf count = 133
\[\left \{\left \{y(x)\to b^{\frac {a-1}{2}} \left (\frac {1}{x}\right )^{\frac {a-1}{2}} \left (c_1 \Gamma \left (1-\sqrt {a^2-2 a-4 c+1}\right ) J_{-\sqrt {a^2-2 a-4 c+1}}\left (2 \sqrt {b} \sqrt {\frac {1}{x}}\right )+c_2 \Gamma \left (\sqrt {a^2-2 a-4 c+1}+1\right ) J_{\sqrt {a^2-2 a-4 c+1}}\left (2 \sqrt {b} \sqrt {\frac {1}{x}}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.064 (sec), leaf count = 71
\[\left [y \left (x \right ) = \textit {\_C1} \,x^{-\frac {a}{2}+\frac {1}{2}} \BesselJ \left (-\sqrt {a^{2}-2 a -4 c +1}, \frac {2 \sqrt {b}}{\sqrt {x}}\right )+\textit {\_C2} \,x^{-\frac {a}{2}+\frac {1}{2}} \BesselY \left (-\sqrt {a^{2}-2 a -4 c +1}, \frac {2 \sqrt {b}}{\sqrt {x}}\right )\right ]\] Mathematica raw input
DSolve[(b + c*x)*y[x] + a*x^2*y'[x] + x^3*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> b^((-1 + a)/2)*(x^(-1))^((-1 + a)/2)*(BesselJ[-Sqrt[1 - 2*a + a^2 - 4*c], 2*Sqrt[b]*Sqrt[x^(-1)]]*C[1]*Gamma[1 - Sqrt[1 - 2*a + a^2 - 4*c]] + BesselJ[Sqrt[1 - 2*a + a^2 - 4*c], 2*Sqrt[b]*Sqrt[x^(-1)]]*C[2]*Gamma[1 + Sqrt[1 - 2*a + a^2 - 4*c]])}}
Maple raw input
dsolve(x^3*diff(diff(y(x),x),x)+a*x^2*diff(y(x),x)+(c*x+b)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*x^(-1/2*a+1/2)*BesselJ(-(a^2-2*a-4*c+1)^(1/2),2*b^(1/2)/x^(1/2))+_C2*x^(-1/2*a+1/2)*BesselY(-(a^2-2*a-4*c+1)^(1/2),2*b^(1/2)/x^(1/2))]