ODE
\[ y(x) (a+b x)+2 a x y'(x)+x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.063338 (sec), leaf count = 123
\[\left \{\left \{y(x)\to b^{\frac {1}{2}-a} x^{\frac {1}{2}-a} \left (c_1 \Gamma \left (1-\sqrt {4 a^2-8 a+1}\right ) J_{-\sqrt {4 a^2-8 a+1}}\left (2 \sqrt {b} \sqrt {x}\right )+c_2 \Gamma \left (\sqrt {4 a^2-8 a+1}+1\right ) J_{\sqrt {4 a^2-8 a+1}}\left (2 \sqrt {b} \sqrt {x}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.02 (sec), leaf count = 59
\[ \left \{ y \left ( x \right ) ={x}^{{\frac {1}{2}}-a} \left ( {{\sl Y}_{\sqrt {4\,{a}^{2}-8\,a+1}}\left (2\,\sqrt {b}\sqrt {x}\right )}{\it \_C2}+{{\sl J}_{\sqrt {4\,{a}^{2}-8\,a+1}}\left (2\,\sqrt {b}\sqrt {x}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[(a + b*x)*y[x] + 2*a*x*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> b^(1/2 - a)*x^(1/2 - a)*(BesselJ[-Sqrt[1 - 8*a + 4*a^2], 2*Sqrt[b]*Sqrt[x]]*C[1]*Gamma[1 - Sqrt[1 - 8*a + 4*a^2]] + BesselJ[Sqrt[1 - 8*a + 4*a^2], 2*Sqrt[b]*Sqrt[x]]*C[2]*Gamma[1 + Sqrt[1 - 8*a + 4*a^2]])}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+2*a*x*diff(y(x),x)+(b*x+a)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = x^(1/2-a)*(BesselY((4*a^2-8*a+1)^(1/2),2*b^(1/2)*x^(1/2))*_C2+BesselJ((4*a^2-8*a+1)^(1/2),2*b^(1/2)*x^(1/2))*_C1)