4.20.1 \(-(a-2 b x) y'(x)-b y(x)+y(x) y'(x)^2=0\)

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

[[_homogeneous, `class C`], _rational, _dAlembert]

Book solution method
Change of variable

Mathematica
cpu = 0.316216 (sec), leaf count = 146

\[\left \{\left \{y(x)\to -\sqrt {2} e^{\frac {c_1}{2}} \sqrt {a+2 b \left (e^{c_1}-x\right )}\right \},\left \{y(x)\to \sqrt {2} e^{\frac {c_1}{2}} \sqrt {a+2 b \left (e^{c_1}-x\right )}\right \},\left \{y(x)\to -\frac {e^{\frac {c_1}{2}} \sqrt {-2 a+4 b x+e^{c_1}}}{2 \sqrt {b}}\right \},\left \{y(x)\to \frac {e^{\frac {c_1}{2}} \sqrt {-2 a+4 b x+e^{c_1}}}{2 \sqrt {b}}\right \}\right \}\]

Maple
cpu = 0.795 (sec), leaf count = 73

\[ \left \{ {\frac {4\,b \left (y \relax (x ) \right ) ^{2}+ \left (-2\,bx+a \right ) ^{2}}{4\,b}}=0,\ln \left (2\,bx-a \right ) -{\it Artanh} \left ({\frac {1}{\sqrt {4\,{\frac {b \left (y \relax (x ) \right ) ^{2}}{ \left (-2\,bx+a \right ) ^{2}}}+1}}} \right ) +\ln \left ({\frac {y \relax (x ) }{-2\,bx+a}} \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> -(Sqrt[2]*E^(C[1]/2)*Sqrt[a + 2*b*(E^C[1] - x)])}, {y[x] -> Sqrt[2]*E^
(C[1]/2)*Sqrt[a + 2*b*(E^C[1] - x)]}, {y[x] -> -(E^(C[1]/2)*Sqrt[-2*a + E^C[1] +
 4*b*x])/(2*Sqrt[b])}, {y[x] -> (E^(C[1]/2)*Sqrt[-2*a + E^C[1] + 4*b*x])/(2*Sqrt
[b])}}

Maple raw input

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

Maple raw output

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