ODE
\[ \text {a1} x y'(x)+y(x) (\text {a2}+\text {b2} x)+x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.225572 (sec), leaf count = 131
\[\left \{\left \{y(x)\to \text {b2}^{\frac {1}{2}-\frac {\text {a1}}{2}} x^{\frac {1}{2}-\frac {\text {a1}}{2}} \left (c_1 \Gamma \left (1-\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}\right ) J_{-\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}}\left (2 \sqrt {\text {b2}} \sqrt {x}\right )+c_2 \Gamma \left (\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}+1\right ) J_{\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}}\left (2 \sqrt {\text {b2}} \sqrt {x}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.028 (sec), leaf count = 67
\[\left [y \left (x \right ) = \textit {\_C1} \,x^{-\frac {\mathit {a1}}{2}+\frac {1}{2}} \BesselJ \left (\sqrt {\mathit {a1}^{2}-2 \mathit {a1} -4 \mathit {a2} +1}, 2 \sqrt {\mathit {b2}}\, \sqrt {x}\right )+\textit {\_C2} \,x^{-\frac {\mathit {a1}}{2}+\frac {1}{2}} \BesselY \left (\sqrt {\mathit {a1}^{2}-2 \mathit {a1} -4 \mathit {a2} +1}, 2 \sqrt {\mathit {b2}}\, \sqrt {x}\right )\right ]\] Mathematica raw input
DSolve[(a2 + b2*x)*y[x] + a1*x*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> b2^(1/2 - a1/2)*x^(1/2 - a1/2)*(BesselJ[-Sqrt[1 - 2*a1 + a1^2 - 4*a2], 2*Sqrt[b2]*Sqrt[x]]*C[1]*Gamma[1 - Sqrt[1 - 2*a1 + a1^2 - 4*a2]] + BesselJ[Sqrt[1 - 2*a1 + a1^2 - 4*a2], 2*Sqrt[b2]*Sqrt[x]]*C[2]*Gamma[1 + Sqrt[1 - 2*a1 + a1^2 - 4*a2]])}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+a1*x*diff(y(x),x)+(b2*x+a2)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*x^(-1/2*a1+1/2)*BesselJ((a1^2-2*a1-4*a2+1)^(1/2),2*b2^(1/2)*x^(1/2))+_C2*x^(-1/2*a1+1/2)*BesselY((a1^2-2*a1-4*a2+1)^(1/2),2*b2^(1/2)*x^(1/2))]