##### 4.18.17 $$x y'(x)^2-(3 x-y(x)) y'(x)+y(x)=0$$

ODE
$x y'(x)^2-(3 x-y(x)) y'(x)+y(x)=0$ ODE Classiﬁcation

[[_homogeneous, class A], _rational, _dAlembert]

Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)

Mathematica
cpu = 2.77566 (sec), leaf count = 371

$\left \{\text {Solve}\left [\frac {1}{8} \left (\frac {(x-y(x)) \sqrt {\frac {9 x-y(x)}{x-y(x)}}}{x}+\sqrt {\frac {y(x)}{x}-9} \sqrt {\frac {y(x)}{x}-1}-3 \log \left (\frac {y(x)}{x}\right )-\frac {10 \sqrt {\frac {y(x)}{x}-9} \sin ^{-1}\left (\frac {\sqrt {9-\frac {y(x)}{x}}}{2 \sqrt {2}}\right )}{\sqrt {9-\frac {y(x)}{x}}}+6 \tanh ^{-1}\left (\frac {1}{3} \sqrt {\frac {9 x-y(x)}{x-y(x)}}\right )+8 \tanh ^{-1}\left (\sqrt {\frac {9 x-y(x)}{x-y(x)}}\right )\right )=\frac {\log (x)}{2}+c_1,y(x)\right ],\text {Solve}\left [\frac {1}{8} \left (\frac {(x-y(x)) \sqrt {\frac {9 x-y(x)}{x-y(x)}}}{x}+\sqrt {\frac {y(x)}{x}-9} \sqrt {\frac {y(x)}{x}-1}+3 \log \left (\frac {y(x)}{x}\right )-\frac {10 \sqrt {\frac {y(x)}{x}-9} \sin ^{-1}\left (\frac {\sqrt {9-\frac {y(x)}{x}}}{2 \sqrt {2}}\right )}{\sqrt {9-\frac {y(x)}{x}}}+6 \tanh ^{-1}\left (\frac {1}{3} \sqrt {\frac {9 x-y(x)}{x-y(x)}}\right )+8 \tanh ^{-1}\left (\sqrt {\frac {9 x-y(x)}{x-y(x)}}\right )\right )+\frac {\log (x)}{2}=c_1,y(x)\right ]\right \}$

Maple
cpu = 0.161 (sec), leaf count = 136

$\left [y \left (x \right ) = x, -\frac {\textit {\_C1} \left (-y \left (x \right )+5 x +\sqrt {9 x^{2}-10 x y \left (x \right )+y \left (x \right )^{2}}\right )}{x \left (\frac {-y \left (x \right )+3 x +\sqrt {9 x^{2}-10 x y \left (x \right )+y \left (x \right )^{2}}}{x}\right )^{\frac {3}{2}}}+x = 0, \frac {\left (y \left (x \right )-5 x +\sqrt {9 x^{2}-10 x y \left (x \right )+y \left (x \right )^{2}}\right ) \textit {\_C1}}{x \left (\frac {-2 y \left (x \right )+6 x -2 \sqrt {9 x^{2}-10 x y \left (x \right )+y \left (x \right )^{2}}}{x}\right )^{\frac {3}{2}}}+x = 0\right ]$ Mathematica raw input

DSolve[y[x] - (3*x - y[x])*y'[x] + x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[(6*ArcTanh[Sqrt[(9*x - y[x])/(x - y[x])]/3] + 8*ArcTanh[Sqrt[(9*x - y[x])
/(x - y[x])]] - 3*Log[y[x]/x] + ((x - y[x])*Sqrt[(9*x - y[x])/(x - y[x])])/x - (
10*ArcSin[Sqrt[9 - y[x]/x]/(2*Sqrt[2])]*Sqrt[-9 + y[x]/x])/Sqrt[9 - y[x]/x] + Sq
rt[-9 + y[x]/x]*Sqrt[-1 + y[x]/x])/8 == C[1] + Log[x]/2, y[x]], Solve[Log[x]/2 +
 (6*ArcTanh[Sqrt[(9*x - y[x])/(x - y[x])]/3] + 8*ArcTanh[Sqrt[(9*x - y[x])/(x -
y[x])]] + 3*Log[y[x]/x] + ((x - y[x])*Sqrt[(9*x - y[x])/(x - y[x])])/x - (10*Arc
Sin[Sqrt[9 - y[x]/x]/(2*Sqrt[2])]*Sqrt[-9 + y[x]/x])/Sqrt[9 - y[x]/x] + Sqrt[-9
+ y[x]/x]*Sqrt[-1 + y[x]/x])/8 == C[1], y[x]]}

Maple raw input

dsolve(x*diff(y(x),x)^2-(3*x-y(x))*diff(y(x),x)+y(x) = 0, y(x))

Maple raw output

[y(x) = x, -_C1*(-y(x)+5*x+(9*x^2-10*x*y(x)+y(x)^2)^(1/2))/x/(1/x*(-y(x)+3*x+(9*
x^2-10*x*y(x)+y(x)^2)^(1/2)))^(3/2)+x = 0, (y(x)-5*x+(9*x^2-10*x*y(x)+y(x)^2)^(1
/2))*_C1/x/((-2*y(x)+6*x-2*(9*x^2-10*x*y(x)+y(x)^2)^(1/2))/x)^(3/2)+x = 0]