Internal problem ID [9248]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1669.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x -x^{2} \left (y^{\prime }\right )^{2}+2 y^{\prime }+y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.059 (sec). Leaf size: 32
dsolve(x*diff(diff(y(x),x),x)-x^2*diff(y(x),x)^2+2*diff(y(x),x)+y(x)^2=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {\RootOf \left (-\ln \relax (x )+c_{2}-\left (\int _{}^{\textit {\_Z}}\frac {1}{{\mathrm e}^{\textit {\_f}} c_{1}-2 \textit {\_f} -1}d \textit {\_f} \right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.342 (sec). Leaf size: 160
DSolve[y[x]^2 + 2*y'[x] - x^2*y'[x]^2 + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {x}{e^{x K[1]} c_1+2 x K[1]+1}dK[1]-\int _1^x\left (\int _1^{y(x)}\left (\frac {\left (e^{K[1] K[2]} c_1 K[1]+2 K[1]\right ) K[2]}{\left (e^{K[1] K[2]} c_1+2 K[1] K[2]+1\right ){}^2}-\frac {1}{e^{K[1] K[2]} c_1+2 K[1] K[2]+1}\right )dK[1]-\frac {e^{K[2] y(x)} c_1+K[2] y(x)+1}{K[2] \left (e^{K[2] y(x)} c_1+2 K[2] y(x)+1\right )}\right )dK[2]=c_2,y(x)\right ] \]