4.38.8 \(a x^2 y'(x)^2+x y''(x)+2 y'(x)=b\)

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 = 51.0929 (sec), leaf count = 115

\[\left \{\left \{y(x)\to c_2+\int _1^x \frac {i \sqrt {b} \left (Y_1\left (-i \sqrt {a} \sqrt {b} K[1]\right )-c_1 J_1\left (i \sqrt {a} \sqrt {b} K[1]\right )\right )}{\sqrt {a} K[1] \left (c_1 J_0\left (i \sqrt {a} \sqrt {b} K[1]\right )+Y_0\left (-i \sqrt {a} \sqrt {b} K[1]\right )\right )} \, dK[1]\right \}\right \}\]

Maple ✓
cpu = 0.813 (sec), leaf count = 70

\[ \left \{ y \left ( x \right ) =\int \!-{\frac {1}{ax}\sqrt {-ab} \left ( {{\sl Y}_{1}\left (\sqrt {-ab}x\right )}{\it \_C1}+{{\sl J}_{1}\left (\sqrt {-ab}x\right )} \right ) \left ( {\it \_C1}\,{{\sl Y}_{0}\left (\sqrt {-ab}x\right )}+{{\sl J}_{0}\left (\sqrt {-ab}x\right )} \right ) ^{-1}}\,{\rm d}x+{\it \_C2} \right \} \] 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] + Integrate[(I*Sqrt[b]*(BesselY[1, (-I)*Sqrt[a]*Sqrt[b]*K[1]] - B
esselJ[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),'implicit')

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
))/x/a/(_C1*BesselY(0,(-a*b)^(1/2)*x)+BesselJ(0,(-a*b)^(1/2)*x)),x)+_C2