Internal problem ID [8724]
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]
\[ \boxed {{y^{\prime }}^{2}-2 y y^{\prime }=2 x} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 223
dsolve(diff(y(x),x)^2-2*y(x)*diff(y(x),x)-2*x = 0,y(x), singsol=all)
\begin{align*} \frac {\frac {\left (y \left (x \right )-\sqrt {y \left (x \right )^{2}+2 x}\right ) \operatorname {arcsinh}\left (-y \left (x \right )+\sqrt {y \left (x \right )^{2}+2 x}\right )}{2}+x \sqrt {2 y \left (x \right )^{2}+2 x -2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+2 x}+1}-2 c_{1} y \left (x \right )+2 c_{1} \sqrt {y \left (x \right )^{2}+2 x}}{\sqrt {2 y \left (x \right )^{2}+2 x -2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+2 x}+1}} &= 0 \\ \frac {\frac {\left (-y \left (x \right )-\sqrt {y \left (x \right )^{2}+2 x}\right ) \operatorname {arcsinh}\left (y \left (x \right )+\sqrt {y \left (x \right )^{2}+2 x}\right )}{2}+x \sqrt {2 y \left (x \right )^{2}+2 x +2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+2 x}+1}+2 c_{1} y \left (x \right )+2 c_{1} \sqrt {y \left (x \right )^{2}+2 x}}{\sqrt {2 y \left (x \right )^{2}+2 x +2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+2 x}+1}} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.713 (sec). Leaf size: 74
DSolve[-2*x - 2*y[x]*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=-\frac {K[1] \log \left (\sqrt {K[1]^2+1}-K[1]\right )}{2 \sqrt {K[1]^2+1}}+\frac {c_1 K[1]}{\sqrt {K[1]^2+1}},y(x)=\frac {K[1]}{2}-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ] \]