ODE
\[ (a-y(x)) y'(x)+b+x y'(x)^2=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]
Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)
Mathematica ✓
cpu = 0.160425 (sec), leaf count = 17
\[\left \{\left \{y(x)\to a+\frac {b}{c_1}+c_1 x\right \}\right \}\]
Maple ✓
cpu = 0.064 (sec), leaf count = 41
\[\left [y \left (x \right ) = a -2 \sqrt {b x}, y \left (x \right ) = a +2 \sqrt {b x}, y \left (x \right ) = x \textit {\_C1} +\frac {\textit {\_C1} a +b}{\textit {\_C1}}\right ]\] Mathematica raw input
DSolve[b + (a - y[x])*y'[x] + x*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> a + b/C[1] + x*C[1]}}
Maple raw input
dsolve(x*diff(y(x),x)^2+(a-y(x))*diff(y(x),x)+b = 0, y(x))
Maple raw output
[y(x) = a-2*(b*x)^(1/2), y(x) = a+2*(b*x)^(1/2), y(x) = x*_C1+(_C1*a+b)/_C1]