4.18.18 $a-by(x)+bx+x{y}^{\prime }(x{)}^2-y(x)=0$

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

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

Book solution method
Clairaut’s equation and related types, $f(y-x{y}^{\prime },y^{\prime })=0$

Mathematica ✓
cpu = 2.83238 (sec), leaf count = 323

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

Maple ✓
cpu = 0.034 (sec), leaf count = 196

$$\{[x\left(\mathit{\_}\mathit{T}\right)=\mathit{\_}\mathit{C}\mathit{1}{\left({(\mathit{\_}\mathit{T}-b)}^{-\frac{b^2}{(b+1)(-1+b)}}\right)}^2{\left({(\mathit{\_}\mathit{T}-b)}^{-\frac{b}{(b+1)(-1+b)}}\right)}^2{\left({(\mathit{\_}\mathit{T}-1)}^{-\frac{b}{(b+1)(-1+b)}}\right)}^{-2}{\left({(\mathit{\_}\mathit{T}-1)}^{-\frac{1}{(b+1)(-1+b)}}\right)}^{-2},y\left(\mathit{\_}\mathit{T}\right)=\frac{({\mathit{\_}\mathit{T}}^2+b)\mathit{\_}\mathit{C}\mathit{1}}{b+1}{\left({(\mathit{\_}\mathit{T}-b)}^{\frac{b^2}{(-b-1)(-1+b)}}\right)}^2{\left({(\mathit{\_}\mathit{T}-b)}^{\frac{b}{(-b-1)(-1+b)}}\right)}^2{\left({(\mathit{\_}\mathit{T}-1)}^{\frac{b}{(-b-1)(-1+b)}}\right)}^{-2}{\left({(\mathit{\_}\mathit{T}-1)}^{\frac{1}{(-b-1)(-1+b)}}\right)}^{-2}+\frac{a}{b+1}]\}$$ Mathematica raw input

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

Mathematica raw output

{Solve[((1 + b)*(-2*b*ArcTanh[(b*Sqrt[x])/Sqrt[-a - b*x + y[x] + b*y[x]]] + Log[
a + (1 + b)*(x - y[x])] - b*Log[a + (1 + b)*(b*x - y[x])] - (2*ArcTan[(Sqrt[x]*S
qrt[-a + (1 + b)*y[x]])/(Sqrt[a - (1 + b)*y[x]]*Sqrt[-a - b*x + (1 + b)*y[x]])]*
Sqrt[-a + (1 + b)*y[x]])/Sqrt[a - (1 + b)*y[x]]))/(1 - b^2) == C[1], y[x]], Solv
e[((1 + b)*(2*b*ArcTanh[(b*Sqrt[x])/Sqrt[-a - b*x + y[x] + b*y[x]]] + Log[a + (1
 + b)*(x - y[x])] - b*Log[a + (1 + b)*(b*x - y[x])] + (2*ArcTan[(Sqrt[x]*Sqrt[-a
 + (1 + b)*y[x]])/(Sqrt[a - (1 + b)*y[x]]*Sqrt[-a - b*x + (1 + b)*y[x]])]*Sqrt[-
a + (1 + b)*y[x]])/Sqrt[a - (1 + b)*y[x]]))/(1 - b^2) == C[1], y[x]]}

Maple raw input

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

Maple raw output

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