Internal problem ID [9373]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1799 (book 6.208).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{3} y^{2} y^{\prime \prime }+\left (x +y\right ) \left (y^{\prime } x -y\right )^{3}=0} \]
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 170
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), singsol=all)
\begin{align*} y \left (x \right ) = 0 \\ y \left (x \right ) = \operatorname {RootOf}\left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {i \sqrt {3}\, \operatorname {BesselY}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) c_{1} \sqrt {\textit {\_f}}+i \sqrt {3}\, \operatorname {BesselJ}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \sqrt {\textit {\_f}}+\operatorname {BesselY}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) c_{1} \sqrt {\textit {\_f}}-2 c_{1} \operatorname {BesselY}\left (1+i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \textit {\_f} +\operatorname {BesselJ}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \sqrt {\textit {\_f}}-2 \operatorname {BesselJ}\left (1+i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \textit {\_f}}{\textit {\_f}^{\frac {3}{2}} \left (\operatorname {BesselY}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) c_{1} +\operatorname {BesselJ}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right )\right )}d \textit {\_f} \right )+2 c_{2} \right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 33.841 (sec). Leaf size: 248
DSolve[(x + y[x])*(-y[x] + x*y'[x])^3 + x^3*y[x]^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\int _1^{\frac {y(x)}{x}}\frac {i \sqrt {3} \sqrt {K[2]} \operatorname {BesselJ}\left (i \sqrt {3},2 \sqrt {K[2]}\right )+\sqrt {K[2]} \operatorname {BesselJ}\left (i \sqrt {3},2 \sqrt {K[2]}\right )-2 \operatorname {BesselJ}\left (1+i \sqrt {3},2 \sqrt {K[2]}\right ) K[2]-2 Y_{1+i \sqrt {3}}\left (2 \sqrt {K[2]}\right ) c_1 K[2]+i \sqrt {3} Y_{i \sqrt {3}}\left (2 \sqrt {K[2]}\right ) c_1 \sqrt {K[2]}+Y_{i \sqrt {3}}\left (2 \sqrt {K[2]}\right ) c_1 \sqrt {K[2]}}{\left (\operatorname {BesselJ}\left (i \sqrt {3},2 \sqrt {K[2]}\right )+Y_{i \sqrt {3}}\left (2 \sqrt {K[2]}\right ) c_1\right ) K[2]^{3/2}}dK[2]-2 \log (x)+2 c_2=0,y(x)\right ] \]