##### 4.18.25 $$-a y(x) y'(x)+b+x y'(x)^2=0$$

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

[[_homogeneous, class G], _rational, _dAlembert]

Book solution method
No Missing Variables ODE, Solve for $$y$$

Mathematica
cpu = 0.741388 (sec), leaf count = 147

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

Maple
cpu = 3.055 (sec), leaf count = 646

$\left [\frac {\textit {\_C1} \left (2 \left (\frac {a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a^{3} y \left (x \right )^{2}+2 \left (\frac {a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} \sqrt {a^{2} y \left (x \right )^{2}-4 b x}\, a^{2} y \left (x \right )-\left (\frac {a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a^{2} y \left (x \right )^{2}-\left (\frac {a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} \sqrt {a^{2} y \left (x \right )^{2}-4 b x}\, a y \left (x \right )-4 \left (\frac {a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a b x +2 \left (\frac {a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} b x \right )}{\left (a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}\right )^{2}}+x -\frac {4 b \,x^{2}}{\left (2 a -1\right ) \left (a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}\right )^{2}} = 0, \frac {\textit {\_C1} \left (-2 \left (-\frac {-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a^{3} y \left (x \right )^{2}+2 \left (-\frac {-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} \sqrt {a^{2} y \left (x \right )^{2}-4 b x}\, a^{2} y \left (x \right )+\left (-\frac {-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a^{2} y \left (x \right )^{2}-\left (-\frac {-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} \sqrt {a^{2} y \left (x \right )^{2}-4 b x}\, a y \left (x \right )+4 \left (-\frac {-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} a b x -2 \left (-\frac {-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{2 x}\right )^{\frac {1}{a -1}} b x \right )}{\left (-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}\right )^{2}}+x -\frac {4 b \,x^{2}}{\left (2 a -1\right ) \left (-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}\right )^{2}} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

{Solve[(2*((-1 + a)*Log[y[x] - a*y[x] - Sqrt[-4*b*x + a^2*y[x]^2]] + a*Log[a*y[x
] - Sqrt[-4*b*x + a^2*y[x]^2]]))/(-1 + 2*a) == C[1], y[x]], Solve[(2*((-1 + a)*L
og[y[x] - a*y[x] + Sqrt[-4*b*x + a^2*y[x]^2]] + a*Log[a*y[x] + Sqrt[-4*b*x + a^2
*y[x]^2]]))/(-1 + 2*a) == C[1], y[x]]}

Maple raw input

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

Maple raw output

[_C1*(2*(1/2/x*(a*y(x)+(a^2*y(x)^2-4*b*x)^(1/2)))^(1/(a-1))*a^3*y(x)^2+2*(1/2/x*
(a*y(x)+(a^2*y(x)^2-4*b*x)^(1/2)))^(1/(a-1))*(a^2*y(x)^2-4*b*x)^(1/2)*a^2*y(x)-(
1/2/x*(a*y(x)+(a^2*y(x)^2-4*b*x)^(1/2)))^(1/(a-1))*a^2*y(x)^2-(1/2/x*(a*y(x)+(a^
2*y(x)^2-4*b*x)^(1/2)))^(1/(a-1))*(a^2*y(x)^2-4*b*x)^(1/2)*a*y(x)-4*(1/2/x*(a*y(
x)+(a^2*y(x)^2-4*b*x)^(1/2)))^(1/(a-1))*a*b*x+2*(1/2/x*(a*y(x)+(a^2*y(x)^2-4*b*x
)^(1/2)))^(1/(a-1))*b*x)/(a*y(x)+(a^2*y(x)^2-4*b*x)^(1/2))^2+x-4*b*x^2/(2*a-1)/(
a*y(x)+(a^2*y(x)^2-4*b*x)^(1/2))^2 = 0, _C1*(-2*(-1/2*(-a*y(x)+(a^2*y(x)^2-4*b*x
)^(1/2))/x)^(1/(a-1))*a^3*y(x)^2+2*(-1/2*(-a*y(x)+(a^2*y(x)^2-4*b*x)^(1/2))/x)^(
1/(a-1))*(a^2*y(x)^2-4*b*x)^(1/2)*a^2*y(x)+(-1/2*(-a*y(x)+(a^2*y(x)^2-4*b*x)^(1/
2))/x)^(1/(a-1))*a^2*y(x)^2-(-1/2*(-a*y(x)+(a^2*y(x)^2-4*b*x)^(1/2))/x)^(1/(a-1)
)*(a^2*y(x)^2-4*b*x)^(1/2)*a*y(x)+4*(-1/2*(-a*y(x)+(a^2*y(x)^2-4*b*x)^(1/2))/x)^
(1/(a-1))*a*b*x-2*(-1/2*(-a*y(x)+(a^2*y(x)^2-4*b*x)^(1/2))/x)^(1/(a-1))*b*x)/(-a
*y(x)+(a^2*y(x)^2-4*b*x)^(1/2))^2+x-4*b*x^2/(2*a-1)/(-a*y(x)+(a^2*y(x)^2-4*b*x)^
(1/2))^2 = 0]