ODE
\[ a x^2 y'(x)^2+x y''(x)+2 y'(x)=b \] ODE Classification
[[_2nd_order, _missing_y]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.478177 (sec), leaf count = 116
\[\left \{\left \{y(x)\to \int _1^x\frac {i \sqrt {b} \left (Y_1\left (-i \sqrt {a} \sqrt {b} K[1]\right )-J_1\left (i \sqrt {a} \sqrt {b} K[1]\right ) c_1\right )}{\sqrt {a} \left (Y_0\left (-i \sqrt {a} \sqrt {b} K[1]\right )+J_0\left (i \sqrt {a} \sqrt {b} K[1]\right ) c_1\right ) K[1]}dK[1]+c_2\right \}\right \}\]
Maple ✓
cpu = 2.523 (sec), leaf count = 70
\[\left [y \left (x \right ) = \int -\frac {\sqrt {-a b}\, \left (\BesselY \left (1, \sqrt {-a b}\, x \right ) \textit {\_C1} +\BesselJ \left (1, \sqrt {-a b}\, x \right )\right )}{a x \left (\textit {\_C1} \BesselY \left (0, \sqrt {-a b}\, x \right )+\BesselJ \left (0, \sqrt {-a b}\, x \right )\right )}d x +\textit {\_C2}\right ]\] Mathematica raw input
DSolve[2*y'[x] + a*x^2*y'[x]^2 + x*y''[x] == b,y[x],x]
Mathematica raw output
{{y[x] -> C[2] + Inactive[Integrate][(I*Sqrt[b]*(BesselY[1, (-I)*Sqrt[a]*Sqrt[b]*K[1]] - BesselJ[1, I*Sqrt[a]*Sqrt[b]*K[1]]*C[1]))/(Sqrt[a]*(BesselY[0, (-I)*Sqrt[a]*Sqrt[b]*K[1]] + BesselJ[0, I*Sqrt[a]*Sqrt[b]*K[1]]*C[1])*K[1]), {K[1], 1, x}]}}
Maple raw input
dsolve(x*diff(diff(y(x),x),x)+a*x^2*diff(y(x),x)^2+2*diff(y(x),x) = b, y(x))
Maple raw output
[y(x) = Int(-(-a*b)^(1/2)*(BesselY(1,(-a*b)^(1/2)*x)*_C1+BesselJ(1,(-a*b)^(1/2)*x))/a/x/(_C1*BesselY(0,(-a*b)^(1/2)*x)+BesselJ(0,(-a*b)^(1/2)*x)),x)+_C2]