ODE
\[ x y'(x)^2-2 y(x)+x=0 \] ODE Classification
[[_homogeneous, `class A`], _rational, _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 0.468896 (sec), leaf count = 95
\[\left \{\text {Solve}\left [-2 \left (\frac {1}{1-\sqrt {\frac {2 y(x)}{x}-1}}+\log \left (1-\sqrt {\frac {2 y(x)}{x}-1}\right )\right )=\log (x)+c_1,y(x)\right ],\text {Solve}\left [2 \left (\frac {1}{\sqrt {\frac {2 y(x)}{x}-1}+1}+\log \left (\sqrt {\frac {2 y(x)}{x}-1}+1\right )\right )+\log (x)=c_1,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.176 (sec), leaf count = 73
\[\left [y \left (x \right ) = \left (\frac {\left (\LambertW \left (\frac {\sqrt {x \textit {\_C1}}}{\textit {\_C1}}\right )+1\right )^{2}}{2 \LambertW \left (\frac {\sqrt {x \textit {\_C1}}}{\textit {\_C1}}\right )^{2}}+\frac {1}{2}\right ) x, y \left (x \right ) = \left (\frac {\left (\LambertW \left (-\frac {\sqrt {x \textit {\_C1}}}{\textit {\_C1}}\right )+1\right )^{2}}{2 \LambertW \left (-\frac {\sqrt {x \textit {\_C1}}}{\textit {\_C1}}\right )^{2}}+\frac {1}{2}\right ) x\right ]\] Mathematica raw input
DSolve[x - 2*y[x] + x*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{Solve[-2*(Log[1 - Sqrt[-1 + (2*y[x])/x]] + (1 - Sqrt[-1 + (2*y[x])/x])^(-1)) == C[1] + Log[x], y[x]], Solve[Log[x] + 2*(Log[1 + Sqrt[-1 + (2*y[x])/x]] + (1 + Sqrt[-1 + (2*y[x])/x])^(-1)) == C[1], y[x]]}
Maple raw input
dsolve(x*diff(y(x),x)^2+x-2*y(x) = 0, y(x))
Maple raw output
[y(x) = (1/2*(LambertW(1/_C1*(x*_C1)^(1/2))+1)^2/LambertW(1/_C1*(x*_C1)^(1/2))^2+1/2)*x, y(x) = (1/2*(LambertW(-1/_C1*(x*_C1)^(1/2))+1)^2/LambertW(-1/_C1*(x*_C1)^(1/2))^2+1/2)*x]