4.18.15 \(x y'(x)^2+(x-y(x)) y'(x)-y(x)+1=0\)

ODE
\[ x y'(x)^2+(x-y(x)) y'(x)-y(x)+1=0 \] ODE Classification

[[_1st_order, _with_linear_symmetries], _rational, _dAlembert]

Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)

Mathematica
cpu = 0.311527 (sec), leaf count = 16

\[\left \{\left \{y(x)\to \left (\frac {1}{c_1}-1\right ) x+c_1\right \}\right \}\]

Maple
cpu = 0.03 (sec), leaf count = 38

\[ \left \{ \left (y \relax (x ) \right ) ^{2}+2\,xy \relax (x ) +{x}^{2}-4\,x=0,y \relax (x ) ={\frac {x{{\it \_C1}}^{2}+{\it \_C1}\,x+1}{{\it \_C1}+1}} \right \} \] Mathematica raw input

DSolve[1 - y[x] + (x - y[x])*y'[x] + x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> x*(-1 + C[1]^(-1)) + C[1]}}

Maple raw input

dsolve(x*diff(y(x),x)^2+(x-y(x))*diff(y(x),x)+1-y(x) = 0, y(x),'implicit')

Maple raw output

y(x)^2+2*x*y(x)+x^2-4*x = 0, y(x) = (_C1^2*x+_C1*x+1)/(_C1+1)