ODE
\[ a \left (x y'(x)-y(x)\right )^2+x^2 y''(x)=b x^2 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 49.7453 (sec), leaf count = 117
\[\left \{\left \{y(x)\to x \left (c_2+\int _1^x \frac {i \sqrt {b} \left (Y_1\left (-i \sqrt {a} \sqrt {b} K[2]\right )-c_1 J_1\left (i \sqrt {a} \sqrt {b} K[2]\right )\right )}{\sqrt {a} K[2] \left (c_1 J_0\left (i \sqrt {a} \sqrt {b} K[2]\right )+Y_0\left (-i \sqrt {a} \sqrt {b} K[2]\right )\right )} \, dK[2]\right )\right \}\right \}\]
Maple ✓
cpu = 0.408 (sec), leaf count = 79
\[ \left \{ {\frac {y \left ( x \right ) }{x}}-\int \!-{\frac {1}{ax}\sqrt {-ab} \left ( {{\sl Y}_{1}\left (\sqrt {-ab}x\right )}{\it \_C1}+{{\sl J}_{1}\left (\sqrt {-ab}x\right )} \right ) \left ( {\it \_C1}\,{{\sl Y}_{0}\left (\sqrt {-ab}x\right )}+{{\sl J}_{0}\left (\sqrt {-ab}x\right )} \right ) ^{-1}}\,{\rm d}x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[a*(-y[x] + x*y'[x])^2 + x^2*y''[x] == b*x^2,y[x],x]
Mathematica raw output
{{y[x] -> x*(C[2] + Integrate[(I*Sqrt[b]*(BesselY[1, (-I)*Sqrt[a]*Sqrt[b]*K[2]] - BesselJ[1, I*Sqrt[a]*Sqrt[b]*K[2]]*C[1]))/(Sqrt[a]*(BesselY[0, (-I)*Sqrt[a]*Sqrt[b]*K[2]] + BesselJ[0, I*Sqrt[a]*Sqrt[b]*K[2]]*C[1])*K[2]), {K[2], 1, x}])}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+a*(x*diff(y(x),x)-y(x))^2 = b*x^2, y(x),'implicit')
Maple raw output
y(x)/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 = 0