ODE
\[ (a+b x) y'(x)+c+y'(x)^2=b y(x) \] ODE Classification
[[_1st_order, _with_linear_symmetries], _Clairaut]
Book solution method
Clairaut’s equation and related types, main form
Mathematica ✓
cpu = 1.78067 (sec), leaf count = 101
\[\left \{\left \{y(x)\to \frac {-a^2-2 a b x+b^2 (2 x+1)-2 b e^{c_1} (x+1)+e^{2 c_1}+4 c}{4 b}\right \},\left \{y(x)\to \frac {-a^2-2 a b x+b^2 (2 x+1)+2 b e^{c_1} (x+1)+e^{2 c_1}+4 c}{4 b}\right \}\right \}\]
Maple ✓
cpu = 0.023 (sec), leaf count = 50
\[ \left \{ y \left ( x \right ) ={\frac {{{\it \_C1}}^{2}+ \left ( bx+a \right ) {\it \_C1}+c}{b}},y \left ( x \right ) ={\frac {-{b}^{2}{x}^{2}-2\,abx-{a}^{2}+4\,c}{4\,b}} \right \} \] Mathematica raw input
DSolve[c + (a + b*x)*y'[x] + y'[x]^2 == b*y[x],y[x],x]
Mathematica raw output
{{y[x] -> (-a^2 + 4*c + E^(2*C[1]) - 2*a*b*x - 2*b*E^C[1]*(1 + x) + b^2*(1 + 2*x))/(4*b)}, {y[x] -> (-a^2 + 4*c + E^(2*C[1]) - 2*a*b*x + 2*b*E^C[1]*(1 + x) + b^2*(1 + 2*x))/(4*b)}}
Maple raw input
dsolve(diff(y(x),x)^2+(b*x+a)*diff(y(x),x)+c = b*y(x), y(x),'implicit')
Maple raw output
y(x) = 1/4*(-b^2*x^2-2*a*b*x-a^2+4*c)/b, y(x) = (_C1^2+(b*x+a)*_C1+c)/b