4.18.18 \(a-b y(x)+b x+x y'(x)^2-y(x)=0\)

ODE
\[ a-b y(x)+b x+x y'(x)^2-y(x)=0 \] ODE Classification

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

Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)

Mathematica ✓
cpu = 2.18616 (sec), leaf count = 320

\[\left \{\text {Solve}\left [\frac {\log (a+(b+1) (x-y(x)))-b \log (a+(b+1) (b x-y(x)))+\frac {2 \sqrt {(b+1) y(x)-a} \tan ^{-1}\left (\frac {\sqrt {x} \sqrt {(b+1) y(x)-a}}{\sqrt {a-(b+1) y(x)} \sqrt {-a+(b+1) y(x)-b x}}\right )}{\sqrt {a-(b+1) y(x)}}+2 b \tanh ^{-1}\left (\frac {b \sqrt {x}}{\sqrt {-a+b y(x)-b x+y(x)}}\right )+(b-1) c_1}{b-1}=0,y(x)\right ],\text {Solve}\left [\frac {(b+1) \left (-\log (a+(b+1) (x-y(x)))+b \log (a+(b+1) (b x-y(x)))+\frac {2 \sqrt {(b+1) y(x)-a} \tan ^{-1}\left (\frac {\sqrt {x} \sqrt {(b+1) y(x)-a}}{\sqrt {a-(b+1) y(x)} \sqrt {-a+(b+1) y(x)-b x}}\right )}{\sqrt {a-(b+1) y(x)}}+2 b \tanh ^{-1}\left (\frac {b \sqrt {x}}{\sqrt {-a+b y(x)-b x+y(x)}}\right )\right )}{b^2-1}=c_1,y(x)\right ]\right \}\]

Maple ✓
cpu = 0.642 (sec), leaf count = 77

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

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

Mathematica raw output

{Solve[(2*b*ArcTanh[(b*Sqrt[x])/Sqrt[-a - b*x + y[x] + b*y[x]]] + (-1 + b)*C[1] 
+ Log[a + (1 + b)*(x - y[x])] - b*Log[a + (1 + b)*(b*x - y[x])] + (2*ArcTan[(Sqr
t[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) == 0, y[x]], Sol
ve[((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))

Maple raw output

[y(x) = -((RootOf(_Z*(x/_C1)^(1/2)-b*(x/_C1)^(1/2)-_Z^(1/b)*(x/_C1)^(1/2/b)+(x/_
C1)^(1/2))+1)^2+b)/(-1-b)*x-a/(-1-b)]