ODE
\[ (a-y(x)+x) y'(x)+x y'(x)^2-y(x)=0 \] 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 = 0.877841 (sec), leaf count = 145
\[\left \{\left \{y(x)\to -\frac {-3 a^2+\sqrt {-a e^{4 c_1} \left (a+e^{4 c_1}+4 x\right ){}^2}-3 a e^{4 c_1}+6 a x+2 e^{4 c_1} x}{2 \left (a+e^{4 c_1}\right )}\right \},\left \{y(x)\to \frac {3 a^2+3 a \left (e^{4 c_1}-2 x\right )+\sqrt {-a e^{4 c_1} \left (a+e^{4 c_1}+4 x\right ){}^2}-2 e^{4 c_1} x}{2 \left (a+e^{4 c_1}\right )}\right \}\right \}\]
Maple ✓
cpu = 0.038 (sec), leaf count = 39
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{2}+ \left ( -2\,a+2\,x \right ) y \left ( x \right ) + \left ( a+x \right ) ^{2}=0,y \left ( x \right ) ={\frac {{\it \_C1}\, \left ( {\it \_C1}\,x+a+x \right ) }{{\it \_C1}+1}} \right \} \] Mathematica raw input
DSolve[-y[x] + (a + x - y[x])*y'[x] + x*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(-3*a^2 - 3*a*E^(4*C[1]) + 6*a*x + 2*E^(4*C[1])*x + Sqrt[-(a*E^(4*C[1])*(a + E^(4*C[1]) + 4*x)^2)])/(2*(a + E^(4*C[1])))}, {y[x] -> (3*a^2 + 3*a*(E^(4*C[1]) - 2*x) - 2*E^(4*C[1])*x + Sqrt[-(a*E^(4*C[1])*(a + E^(4*C[1]) + 4*x)^2)])/(2*(a + E^(4*C[1])))}}
Maple raw input
dsolve(x*diff(y(x),x)^2+(a+x-y(x))*diff(y(x),x)-y(x) = 0, y(x),'implicit')
Maple raw output
y(x)^2+(-2*a+2*x)*y(x)+(a+x)^2 = 0, y(x) = _C1*(_C1*x+a+x)/(_C1+1)