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 = 0.154824 (sec), leaf count = 16

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

Maple ✓
cpu = 0.044 (sec), leaf count = 56

\[\left [y \left (x \right ) = -x -2 \sqrt {x}, y \left (x \right ) = -x +2 \sqrt {x}, y \left (x \right ) = \frac {\left (-\textit {\_C1}^{2}-\textit {\_C1} \right ) x}{-\textit {\_C1} -1}-\frac {1}{-\textit {\_C1} -1}\right ]\] 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*C[1] + (1 + C[1])^(-1)}}

Maple raw input

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

Maple raw output

[y(x) = -x-2*x^(1/2), y(x) = -x+2*x^(1/2), y(x) = (-_C1^2-_C1)/(-_C1-1)*x-1/(-_C
1-1)]