4.18.26 $ay(x)y^{\prime }(x)+bx+x{y}^{\prime }(x{)}^2=0$

ODE
$$ay(x)y^{\prime }(x)+bx+x{y}^{\prime }(x{)}^2=0$$ ODE Classification

[[_homogeneous, `class A`], _dAlembert]

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

Mathematica ✓
cpu = 0.591939 (sec), leaf count = 223

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

Maple ✓
cpu = 0.044 (sec), leaf count = 96

$$\{[x\left(\mathit{\_}\mathit{T}\right)=\frac{\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}}{{((1+a){\mathit{\_}\mathit{T}}^2+b)}^{{(1+a)}^{-1}}}{((1+a){\mathit{\_}\mathit{T}}^2+b)}^{-\frac{a}{2+2a}},y\left(\mathit{\_}\mathit{T}\right)=-\frac{({\mathit{\_}\mathit{T}}^2+b)\mathit{\_}\mathit{C}\mathit{1}}{a{((1+a){\mathit{\_}\mathit{T}}^2+b)}^{{(1+a)}^{-1}}}{((1+a){\mathit{\_}\mathit{T}}^2+b)}^{-\frac{a}{2+2a}}]\}$$ 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),'implicit')

Maple raw output

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