##### 4.18.26 $$a y(x) y'(x)+b x+x y'(x)^2=0$$

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

[[_homogeneous, class A], _dAlembert]

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

Mathematica
cpu = 1.07167 (sec), leaf count = 223

$\left \{\text {Solve}\left [\frac {-2 a \tan ^{-1}\left (\frac {a y(x)}{x \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}\right )+(a+2) \left (2 \tan ^{-1}\left (\frac {(a+2) y(x)}{x \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}\right )-i \log \left (\frac {(a+1) y(x)^2}{x^2}+b\right )\right )}{8 (a+1)}=\frac {1}{2} i \log (x)+c_1,y(x)\right ],\text {Solve}\left [\frac {-2 a \tan ^{-1}\left (\frac {a y(x)}{x \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}\right )+(a+2) \left (2 \tan ^{-1}\left (\frac {(a+2) y(x)}{x \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}\right )+i \log \left (\frac {(a+1) y(x)^2}{x^2}+b\right )\right )}{8 (a+1)}=c_1-\frac {1}{2} i \log (x),y(x)\right ]\right \}$

Maple
cpu = 0.455 (sec), leaf count = 224

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

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

Mathematica raw output

{Solve[(-2*a*ArcTan[(a*y[x])/(x*Sqrt[4*b - (a^2*y[x]^2)/x^2])] + (2 + a)*(2*ArcT
an[((2 + a)*y[x])/(x*Sqrt[4*b - (a^2*y[x]^2)/x^2])] - I*Log[b + ((1 + a)*y[x]^2)
/x^2]))/(8*(1 + a)) == C[1] + (I/2)*Log[x], y[x]], Solve[(-2*a*ArcTan[(a*y[x])/(
x*Sqrt[4*b - (a^2*y[x]^2)/x^2])] + (2 + a)*(2*ArcTan[((2 + a)*y[x])/(x*Sqrt[4*b
- (a^2*y[x]^2)/x^2])] + I*Log[b + ((1 + a)*y[x]^2)/x^2]))/(8*(1 + a)) == C[1] -
(I/2)*Log[x], y[x]]}

Maple raw input

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

Maple raw output

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