Internal problem ID [7989]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 409.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational, _dAlembert]
Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{2}-2 y^{\prime }-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.205 (sec). Leaf size: 63
dsolve(x*diff(y(x),x)^2-2*diff(y(x),x)-y(x) = 0,y(x), singsol=all)
\[ y \relax (x ) = x \,{\mathrm e}^{2 \RootOf \left (-x \,{\mathrm e}^{2 \textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}} x +2 \,{\mathrm e}^{\textit {\_Z}}+c_{1}-2 \textit {\_Z} -x \right )}-2 \,{\mathrm e}^{\RootOf \left (-x \,{\mathrm e}^{2 \textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}} x +2 \,{\mathrm e}^{\textit {\_Z}}+c_{1}-2 \textit {\_Z} -x \right )} \]
✓ Solution by Mathematica
Time used: 1.405 (sec). Leaf size: 50
DSolve[-y[x] - 2*y'[x] + x*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=\frac {2 K[1]-2 \log (K[1])}{(K[1]-1)^2}+\frac {c_1}{(K[1]-1)^2},y(x)=x K[1]^2-2 K[1]\right \},\{y(x),K[1]\}\right ] \]