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 = 2.83238 (sec), leaf count = 323

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

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

\[ \left \{ [x \left ({\it \_T} \right ) ={{\it \_C1} \left (\left ({\it \_T}-b \right ) ^{-{\frac {{b}^{2}}{ \left (b+1 \right ) \left (-1+b \right ) }}} \right ) ^{2} \left (\left ({\it \_T}-b \right ) ^{-{\frac {b}{ \left (b+1 \right ) \left (-1+b \right ) }}} \right ) ^{2} \left (\left ({\it \_T}-1 \right ) ^{-{\frac {b}{ \left (b+1 \right ) \left (-1+b \right ) }}} \right ) ^{-2} \left (\left ({\it \_T}-1 \right ) ^{-{\frac {1}{ \left (b+1 \right ) \left (-1+b \right ) }}} \right ) ^{-2}},y \left ({\it \_T} \right ) ={\frac { \left ({{\it \_T}}^{2}+b \right ) {\it \_C1}}{b+1} \left (\left ({\it \_T}-b \right ) ^{{\frac {{b}^{2}}{ \left (-b-1 \right ) \left (-1+b \right ) }}} \right ) ^{2} \left (\left ({\it \_T}-b \right ) ^{{\frac {b}{ \left (-b-1 \right ) \left (-1+b \right ) }}} \right ) ^{2} \left (\left ({\it \_T}-1 \right ) ^{{\frac {b}{ \left (-b-1 \right ) \left (-1+b \right ) }}} \right ) ^{-2} \left (\left ({\it \_T}-1 \right ) ^{{\frac {1}{ \left (-b-1 \right ) \left (-1+b \right ) }}} \right ) ^{-2}}+{\frac {a}{b+1}}] \right \} \] Mathematica raw input

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

Mathematica raw output

{Solve[((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)]