Internal problem ID [7987]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 408.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {x {y^{\prime }}^{2}-2 y+x=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 73
dsolve(x*diff(y(x),x)^2-2*y(x)+x = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \left (\frac {\left (\operatorname {LambertW}\left (\frac {\sqrt {x c_{1}}}{c_{1}}\right )+1\right )^{2}}{2 \operatorname {LambertW}\left (\frac {\sqrt {x c_{1}}}{c_{1}}\right )^{2}}+\frac {1}{2}\right ) x \\ y \left (x \right ) = \left (\frac {\left (\operatorname {LambertW}\left (-\frac {\sqrt {x c_{1}}}{c_{1}}\right )+1\right )^{2}}{2 \operatorname {LambertW}\left (-\frac {\sqrt {x c_{1}}}{c_{1}}\right )^{2}}+\frac {1}{2}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.62 (sec). Leaf size: 97
DSolve[x - 2*y[x] + x*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {2}{\sqrt {\frac {2 y(x)}{x}-1}-1}-2 \log \left (\sqrt {\frac {2 y(x)}{x}-1}-1\right )=\log (x)+c_1,y(x)\right ] \\ \text {Solve}\left [\frac {2}{\sqrt {\frac {2 y(x)}{x}-1}+1}+2 \log \left (\sqrt {\frac {2 y(x)}{x}-1}+1\right )=-\log (x)+c_1,y(x)\right ] \\ \end{align*}