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 = 0.484059 (sec), leaf count = 118
\[\left \{\left \{y(x)\to x \left (\int _1^x\frac {i \sqrt {b} \left (Y_1\left (-i \sqrt {a} \sqrt {b} K[2]\right )-J_1\left (i \sqrt {a} \sqrt {b} K[2]\right ) c_1\right )}{\sqrt {a} \left (Y_0\left (-i \sqrt {a} \sqrt {b} K[2]\right )+J_0\left (i \sqrt {a} \sqrt {b} K[2]\right ) c_1\right ) K[2]}dK[2]+c_2\right )\right \}\right \}\]
Maple ✓
cpu = 1.041 (sec), leaf count = 110
\[\left [y \left (x \right ) = \left (\int \left (-\frac {\sqrt {-a b}\, \textit {\_C1} \BesselY \left (1, \sqrt {-a b}\, x \right )}{a x \left (\textit {\_C1} \BesselY \left (0, \sqrt {-a b}\, x \right )+\BesselJ \left (0, \sqrt {-a b}\, x \right )\right )}-\frac {\BesselJ \left (1, \sqrt {-a b}\, x \right ) \sqrt {-a b}}{a x \left (\textit {\_C1} \BesselY \left (0, \sqrt {-a b}\, x \right )+\BesselJ \left (0, \sqrt {-a b}\, x \right )\right )}\right )d x +\textit {\_C2} \right ) x\right ]\] 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] + Inactive[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))
Maple raw output
[y(x) = (Int(-(-a*b)^(1/2)*_C1/a/x/(_C1*BesselY(0,(-a*b)^(1/2)*x)+BesselJ(0,(-a*b)^(1/2)*x))*BesselY(1,(-a*b)^(1/2)*x)-BesselJ(1,(-a*b)^(1/2)*x)*(-a*b)^(1/2)/a/x/(_C1*BesselY(0,(-a*b)^(1/2)*x)+BesselJ(0,(-a*b)^(1/2)*x)),x)+_C2)*x]