Internal problem ID [7968]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 388.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_dAlembert]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}-2 y y^{\prime }-2 x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.224 (sec). Leaf size: 217
dsolve(diff(y(x),x)^2-2*y(x)*diff(y(x),x)-2*x = 0,y(x), singsol=all)
\begin{align*} \frac {\left (-2 y \relax (x )+2 \sqrt {y \relax (x )^{2}+2 x}\right ) c_{1}}{\sqrt {2 y \relax (x )^{2}+2 x -2 y \relax (x ) \sqrt {y \relax (x )^{2}+2 x}+1}}+x +\frac {\left (-y \relax (x )+\sqrt {y \relax (x )^{2}+2 x}\right ) \arcsinh \left (y \relax (x )-\sqrt {y \relax (x )^{2}+2 x}\right )}{2 \sqrt {2 y \relax (x )^{2}+2 x -2 y \relax (x ) \sqrt {y \relax (x )^{2}+2 x}+1}} = 0 \\ \frac {\left (2 y \relax (x )+2 \sqrt {y \relax (x )^{2}+2 x}\right ) c_{1}}{\sqrt {2 y \relax (x )^{2}+2 x +2 y \relax (x ) \sqrt {y \relax (x )^{2}+2 x}+1}}+x -\frac {\left (y \relax (x )+\sqrt {y \relax (x )^{2}+2 x}\right ) \arcsinh \left (y \relax (x )+\sqrt {y \relax (x )^{2}+2 x}\right )}{2 \sqrt {2 y \relax (x )^{2}+2 x +2 y \relax (x ) \sqrt {y \relax (x )^{2}+2 x}+1}} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.332 (sec). Leaf size: 72
DSolve[-2*x - 2*y[x]*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=\frac {K[1] \tanh ^{-1}\left (\frac {K[1]}{\sqrt {K[1]^2+1}}\right )}{2 \sqrt {K[1]^2+1}}+\frac {c_1 K[1]}{\sqrt {K[1]^2+1}},y(x)=\frac {K[1]}{2}-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ] \]