1.407 problem 408

Internal problem ID [7988]

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]

Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{2}-2 y+x=0} \end {gather*}

Solution by Maple

Time used: 0.229 (sec). Leaf size: 73

dsolve(x*diff(y(x),x)^2-2*y(x)+x = 0,y(x), singsol=all)
 

\begin{align*} y \relax (x ) = \left (\frac {\left (\LambertW \left (\frac {\sqrt {c_{1} x}}{c_{1}}\right )+1\right )^{2}}{2 \LambertW \left (\frac {\sqrt {c_{1} x}}{c_{1}}\right )^{2}}+\frac {1}{2}\right ) x \\ y \relax (x ) = \left (\frac {\left (\LambertW \left (-\frac {\sqrt {c_{1} x}}{c_{1}}\right )+1\right )^{2}}{2 \LambertW \left (-\frac {\sqrt {c_{1} x}}{c_{1}}\right )^{2}}+\frac {1}{2}\right ) x \\ \end{align*}

Solution by Mathematica

Time used: 0.57 (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*}