ODE
\[ x^3 y(x)^2 y''(x)+(y(x)+x) \left (x y'(x)-y(x)\right )^3=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 36.6392 (sec), leaf count = 187
\[\text {Solve}\left [2 c_2=2 \log (x)+\int _1^{\frac {y(x)}{x}}\frac {\left (1+i \sqrt {3}\right ) J_{i \sqrt {3}}\left (2 \sqrt {K[2]}\right )+\left (1+i \sqrt {3}\right ) Y_{i \sqrt {3}}\left (2 \sqrt {K[2]}\right ) c_1-2 \left (J_{1+i \sqrt {3}}\left (2 \sqrt {K[2]}\right )+Y_{1+i \sqrt {3}}\left (2 \sqrt {K[2]}\right ) c_1\right ) \sqrt {K[2]}}{\left (J_{i \sqrt {3}}\left (2 \sqrt {K[2]}\right )+Y_{i \sqrt {3}}\left (2 \sqrt {K[2]}\right ) c_1\right ) K[2]}dK[2],y(x)\right ]\]
Maple ✓
cpu = 0.611 (sec), leaf count = 166
\[\left [y \left (x \right ) = \RootOf \left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {i \BesselY \left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \sqrt {3}\, \textit {\_C1} \sqrt {\textit {\_f}}+i \sqrt {3}\, \BesselJ \left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \sqrt {\textit {\_f}}+\BesselY \left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \textit {\_C1} \sqrt {\textit {\_f}}-2 \textit {\_C1} \BesselY \left (1+i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \textit {\_f} +\BesselJ \left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \sqrt {\textit {\_f}}-2 \BesselJ \left (1+i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \textit {\_f}}{\textit {\_f}^{\frac {3}{2}} \left (\BesselY \left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \textit {\_C1} +\BesselJ \left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right )\right )}d \textit {\_f} \right )+2 \textit {\_C2} \right ) x\right ]\] Mathematica raw input
DSolve[(x + y[x])*(-y[x] + x*y'[x])^3 + x^3*y[x]^2*y''[x] == 0,y[x],x]
Mathematica raw output
Solve[2*C[2] == 2*Log[x] + Inactive[Integrate][((1 + I*Sqrt[3])*BesselJ[I*Sqrt[3], 2*Sqrt[K[2]]] + (1 + I*Sqrt[3])*BesselY[I*Sqrt[3], 2*Sqrt[K[2]]]*C[1] - 2*(BesselJ[1 + I*Sqrt[3], 2*Sqrt[K[2]]] + BesselY[1 + I*Sqrt[3], 2*Sqrt[K[2]]]*C[1])*Sqrt[K[2]])/((BesselJ[I*Sqrt[3], 2*Sqrt[K[2]]] + BesselY[I*Sqrt[3], 2*Sqrt[K[2]]]*C[1])*K[2]), {K[2], 1, y[x]/x}], y[x]]
Maple raw input
dsolve(x^3*y(x)^2*diff(diff(y(x),x),x)+(x+y(x))*(x*diff(y(x),x)-y(x))^3 = 0, y(x))
Maple raw output
[y(x) = RootOf(-2*ln(x)-Intat((I*BesselY(I*3^(1/2),2*_f^(1/2))*3^(1/2)*_C1*_f^(1/2)+I*3^(1/2)*BesselJ(I*3^(1/2),2*_f^(1/2))*_f^(1/2)+BesselY(I*3^(1/2),2*_f^(1/2))*_C1*_f^(1/2)-2*_C1*BesselY(1+I*3^(1/2),2*_f^(1/2))*_f+BesselJ(I*3^(1/2),2*_f^(1/2))*_f^(1/2)-2*BesselJ(1+I*3^(1/2),2*_f^(1/2))*_f)/_f^(3/2)/(BesselY(I*3^(1/2),2*_f^(1/2))*_C1+BesselJ(I*3^(1/2),2*_f^(1/2))),_f = _Z)+2*_C2)*x]