##### 4.16.33 $$a y'(x)+b y(x)+y'(x)^2=0$$

ODE
$a y'(x)+b y(x)+y'(x)^2=0$ ODE Classiﬁcation

[_quadrature]

Book solution method
Missing Variables ODE, Independent variable missing, Solve for $$y$$

Mathematica
cpu = 0.397083 (sec), leaf count = 110

$\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {a^2-4 \text {\#1} b}+a \log \left (a-\sqrt {a^2-4 \text {\#1} b}\right )}{2 b}\& \right ]\left [\frac {x}{2}+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {a^2-4 \text {\#1} b}-a \log \left (\sqrt {a^2-4 \text {\#1} b}+a\right )}{2 b}\& \right ]\left [-\frac {x}{2}+c_1\right ]\right \}\right \}$

Maple
cpu = 1.028 (sec), leaf count = 275

$\left [y \left (x \right ) = -\frac {a^{2} \left (\LambertW \left (-\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {\textit {\_C1} b}{a}} {\mathrm e}^{-\frac {b x}{a}} {\mathrm e}^{-1}}{a}\right )+2\right ) \LambertW \left (-\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {\textit {\_C1} b}{a}} {\mathrm e}^{-\frac {b x}{a}} {\mathrm e}^{-1}}{a}\right )}{4 b}, y \left (x \right ) = -\frac {a^{2} \left (\LambertW \left (\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {\textit {\_C1} b}{a}} {\mathrm e}^{-\frac {b x}{a}} {\mathrm e}^{-1}}{a}\right )+2\right ) \LambertW \left (\frac {2 \sqrt {-b}\, {\mathrm e}^{\frac {\textit {\_C1} b}{a}} {\mathrm e}^{-\frac {b x}{a}} {\mathrm e}^{-1}}{a}\right )}{4 b}, y \left (x \right ) = -\frac {{\mathrm e}^{-\frac {a \ln \left (-\frac {1}{4 b}\right )+2 a \LambertW \left (\frac {2 \,{\mathrm e}^{\frac {\textit {\_C1} b}{a}} {\mathrm e}^{-\frac {b x}{a}} {\mathrm e}^{-1}}{a \sqrt {-\frac {1}{b}}}\right )-2 \textit {\_C1} b +2 b x +2 a}{2 a}} \left ({\mathrm e}^{-\frac {a \ln \left (-\frac {1}{4 b}\right )+2 a \LambertW \left (\frac {2 \,{\mathrm e}^{\frac {\textit {\_C1} b}{a}} {\mathrm e}^{-\frac {b x}{a}} {\mathrm e}^{-1}}{a \sqrt {-\frac {1}{b}}}\right )-2 \textit {\_C1} b +2 b x +2 a}{2 a}}+2 a \right )}{4 b}\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[y(x) = -1/4*a^2*(LambertW(-2*(-b)^(1/2)/a*exp(1/a*_C1*b)/exp(1/a*b*x)*exp(-1))+
2)*LambertW(-2*(-b)^(1/2)/a*exp(1/a*_C1*b)/exp(1/a*b*x)*exp(-1))/b, y(x) = -1/4*
a^2*(LambertW(2*(-b)^(1/2)/a*exp(1/a*_C1*b)/exp(1/a*b*x)*exp(-1))+2)*LambertW(2*
(-b)^(1/2)/a*exp(1/a*_C1*b)/exp(1/a*b*x)*exp(-1))/b, y(x) = -1/4*exp(-1/2*(a*ln(
-1/4/b)+2*a*LambertW(2/a/(-1/b)^(1/2)*exp(1/a*_C1*b)/exp(1/a*b*x)*exp(-1))-2*_C1
*b+2*b*x+2*a)/a)*(exp(-1/2*(a*ln(-1/4/b)+2*a*LambertW(2/a/(-1/b)^(1/2)*exp(1/a*_
C1*b)/exp(1/a*b*x)*exp(-1))-2*_C1*b+2*b*x+2*a)/a)+2*a)/b]