ODE
\[ \text {a1} x y'(x)+y(x) \left (\text {a2}+\text {b2} x^2\right )+x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.178407 (sec), leaf count = 78
\[\left \{\left \{y(x)\to x^{\frac {1}{2}-\frac {\text {a1}}{2}} \left (c_1 J_{\frac {1}{2} \sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}}\left (\sqrt {\text {b2}} x\right )+c_2 Y_{\frac {1}{2} \sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}}\left (\sqrt {\text {b2}} x\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.051 (sec), leaf count = 65
\[\left [y \left (x \right ) = \textit {\_C1} \,x^{-\frac {\mathit {a1}}{2}+\frac {1}{2}} \BesselJ \left (\frac {\sqrt {\mathit {a1}^{2}-2 \mathit {a1} -4 \mathit {a2} +1}}{2}, \sqrt {\mathit {b2}}\, x \right )+\textit {\_C2} \,x^{-\frac {\mathit {a1}}{2}+\frac {1}{2}} \BesselY \left (\frac {\sqrt {\mathit {a1}^{2}-2 \mathit {a1} -4 \mathit {a2} +1}}{2}, \sqrt {\mathit {b2}}\, x \right )\right ]\] Mathematica raw input
DSolve[(a2 + b2*x^2)*y[x] + a1*x*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> x^(1/2 - a1/2)*(BesselJ[Sqrt[1 - 2*a1 + a1^2 - 4*a2]/2, Sqrt[b2]*x]*C[1] + BesselY[Sqrt[1 - 2*a1 + a1^2 - 4*a2]/2, Sqrt[b2]*x]*C[2])}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+a1*x*diff(y(x),x)+(b2*x^2+a2)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*x^(-1/2*a1+1/2)*BesselJ(1/2*(a1^2-2*a1-4*a2+1)^(1/2),b2^(1/2)*x)+_C2*x^(-1/2*a1+1/2)*BesselY(1/2*(a1^2-2*a1-4*a2+1)^(1/2),b2^(1/2)*x)]