4.18.5 \(x y'(x)^2+x y'(x)-y(x)=0\)

ODE
\[ x y'(x)^2+x y'(x)-y(x)=0 \] ODE Classification

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
No Missing Variables ODE, Solve for \(y\)

Mathematica ✓
cpu = 1.03203 (sec), leaf count = 180

\[\left \{\text {Solve}\left [\frac {x \left (\sqrt {\frac {4 y(x)}{x}+1}-1\right ) \left (\left (\sqrt {\frac {4 y(x)}{x}+1}-1\right ) \log \left (\sqrt {\frac {4 y(x)}{x}+1}-1\right )-1\right )}{2 x \left (\sqrt {\frac {4 y(x)}{x}+1}-1\right )-4 y(x)}=c_1+\frac {\log (x)}{2},y(x)\right ],\text {Solve}\left [\frac {x \left (\sqrt {\frac {4 y(x)}{x}+1}+1\right ) \left (\left (\sqrt {\frac {4 y(x)}{x}+1}+1\right ) \log \left (\sqrt {\frac {4 y(x)}{x}+1}+1\right )+1\right )}{2 \left (x \sqrt {\frac {4 y(x)}{x}+1}+2 y(x)+x\right )}+\frac {\log (x)}{2}=c_1,y(x)\right ]\right \}\]

Maple ✓
cpu = 0.021 (sec), leaf count = 29

\[ \left \{ [x \left ( {\it \_T} \right ) ={\frac {{{\rm e}^{{{\it \_T}}^{-1}}}{\it \_C1}}{{{\it \_T}}^{2}}},y \left ( {\it \_T} \right ) ={\frac { \left ( {\it \_T}+1 \right ) {{\rm e}^{{{\it \_T}}^{-1}}}{\it \_C1}}{{\it \_T}}}] \right \} \] Mathematica raw input

DSolve[-y[x] + x*y'[x] + x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[(x*(-1 + Sqrt[1 + (4*y[x])/x])*(-1 + Log[-1 + Sqrt[1 + (4*y[x])/x]]*(-1 +
 Sqrt[1 + (4*y[x])/x])))/(-4*y[x] + 2*x*(-1 + Sqrt[1 + (4*y[x])/x])) == C[1] + L
og[x]/2, y[x]], Solve[Log[x]/2 + (x*(1 + Sqrt[1 + (4*y[x])/x])*(1 + Log[1 + Sqrt
[1 + (4*y[x])/x]]*(1 + Sqrt[1 + (4*y[x])/x])))/(2*(x + 2*y[x] + x*Sqrt[1 + (4*y[
x])/x])) == C[1], y[x]]}

Maple raw input

dsolve(x*diff(y(x),x)^2+x*diff(y(x),x)-y(x) = 0, y(x),'implicit')

Maple raw output

[x(_T) = exp(1/_T)/_T^2*_C1, y(_T) = 1/_T*(_T+1)*exp(1/_T)*_C1]