##### 4.17.40 $$x^2+4 x y'(x)+3 y'(x)^2-y(x)=0$$

ODE
$x^2+4 x y'(x)+3 y'(x)^2-y(x)=0$ ODE Classiﬁcation

[[_homogeneous, class G]]

Book solution method
Change of variable

Mathematica
cpu = 7.21413 (sec), leaf count = 796

$\left \{\text {Solve}\left [\frac {\sqrt {1-9 y(x)} \left (\log \left (-3 x^2+2 x-12 y(x)+1\right )-2 c_1\right )+\left (\frac {9 \tanh ^{-1}\left (\frac {-2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right )}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5}}-\frac {9 \tanh ^{-1}\left (\frac {2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right )}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5}}\right ) y(x)+\frac {\tanh ^{-1}\left (\frac {-2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right ) \left (2 \sqrt {1-9 y(x)}-1\right )}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5}}+\frac {\tanh ^{-1}\left (\frac {2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right ) \left (2 \sqrt {1-9 y(x)}+1\right )}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5}}}{2 \sqrt {1-9 y(x)}}=0,y(x)\right ],\text {Solve}\left [\frac {\sqrt {1-9 y(x)} \left (\log \left (-3 x^2+2 x-12 y(x)+1\right )-2 c_1\right )+\left (\frac {9 \tanh ^{-1}\left (\frac {2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right )}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5}}-\frac {9 \tanh ^{-1}\left (\frac {-2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right )}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5}}\right ) y(x)+\frac {\tanh ^{-1}\left (\frac {-2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right ) \left (1-2 \sqrt {1-9 y(x)}\right )}{\sqrt {-9 y(x)-4 \sqrt {1-9 y(x)}+5}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt {1-9 y(x)} x+x+9 y(x)}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5} \sqrt {x^2+3 y(x)}}\right ) \left (2 \sqrt {1-9 y(x)}+1\right )}{\sqrt {-9 y(x)+4 \sqrt {1-9 y(x)}+5}}}{2 \sqrt {1-9 y(x)}}=0,y(x)\right ]\right \}$

Maple
cpu = 0.361 (sec), leaf count = 117

$\left [y \left (x \right ) = -\frac {x^{2}}{3}, y \left (x \right ) = -\frac {5 \textit {\_C1}^{2} x^{2}+2 \textit {\_C1} x \left (-x \textit {\_C1} +\sqrt {3}\right )-3}{12 \textit {\_C1}^{2}}, y \left (x \right ) = -\frac {5 \textit {\_C1}^{2} x^{2}-2 \textit {\_C1} x \left (x \textit {\_C1} +\sqrt {3}\right )-3}{12 \textit {\_C1}^{2}}, y \left (x \right ) = -\frac {5 x^{2}}{12}+\frac {x \left (x -\sqrt {3}\, \textit {\_C1} \right )}{6}+\frac {\textit {\_C1}^{2}}{4}, y \left (x \right ) = -\frac {5 x^{2}}{12}+\frac {x \left (x +\sqrt {3}\, \textit {\_C1} \right )}{6}+\frac {\textit {\_C1}^{2}}{4}\right ]$ Mathematica raw input

DSolve[x^2 - y[x] + 4*x*y'[x] + 3*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[((-2*C[1] + Log[1 + 2*x - 3*x^2 - 12*y[x]])*Sqrt[1 - 9*y[x]] + (ArcTanh[(
x - 2*x*Sqrt[1 - 9*y[x]] + 9*y[x])/(Sqrt[5 - 4*Sqrt[1 - 9*y[x]] - 9*y[x]]*Sqrt[x
^2 + 3*y[x]])]*(-1 + 2*Sqrt[1 - 9*y[x]]))/Sqrt[5 - 4*Sqrt[1 - 9*y[x]] - 9*y[x]]
+ (ArcTanh[(x + 2*x*Sqrt[1 - 9*y[x]] + 9*y[x])/(Sqrt[5 + 4*Sqrt[1 - 9*y[x]] - 9*
y[x]]*Sqrt[x^2 + 3*y[x]])]*(1 + 2*Sqrt[1 - 9*y[x]]))/Sqrt[5 + 4*Sqrt[1 - 9*y[x]]
 - 9*y[x]] + ((9*ArcTanh[(x - 2*x*Sqrt[1 - 9*y[x]] + 9*y[x])/(Sqrt[5 - 4*Sqrt[1
- 9*y[x]] - 9*y[x]]*Sqrt[x^2 + 3*y[x]])])/Sqrt[5 - 4*Sqrt[1 - 9*y[x]] - 9*y[x]]
- (9*ArcTanh[(x + 2*x*Sqrt[1 - 9*y[x]] + 9*y[x])/(Sqrt[5 + 4*Sqrt[1 - 9*y[x]] -
9*y[x]]*Sqrt[x^2 + 3*y[x]])])/Sqrt[5 + 4*Sqrt[1 - 9*y[x]] - 9*y[x]])*y[x])/(2*Sq
rt[1 - 9*y[x]]) == 0, y[x]], Solve[((-2*C[1] + Log[1 + 2*x - 3*x^2 - 12*y[x]])*S
qrt[1 - 9*y[x]] + (ArcTanh[(x - 2*x*Sqrt[1 - 9*y[x]] + 9*y[x])/(Sqrt[5 - 4*Sqrt[
1 - 9*y[x]] - 9*y[x]]*Sqrt[x^2 + 3*y[x]])]*(1 - 2*Sqrt[1 - 9*y[x]]))/Sqrt[5 - 4*
Sqrt[1 - 9*y[x]] - 9*y[x]] - (ArcTanh[(x + 2*x*Sqrt[1 - 9*y[x]] + 9*y[x])/(Sqrt[
5 + 4*Sqrt[1 - 9*y[x]] - 9*y[x]]*Sqrt[x^2 + 3*y[x]])]*(1 + 2*Sqrt[1 - 9*y[x]]))/
Sqrt[5 + 4*Sqrt[1 - 9*y[x]] - 9*y[x]] + ((-9*ArcTanh[(x - 2*x*Sqrt[1 - 9*y[x]] +
 9*y[x])/(Sqrt[5 - 4*Sqrt[1 - 9*y[x]] - 9*y[x]]*Sqrt[x^2 + 3*y[x]])])/Sqrt[5 - 4
*Sqrt[1 - 9*y[x]] - 9*y[x]] + (9*ArcTanh[(x + 2*x*Sqrt[1 - 9*y[x]] + 9*y[x])/(Sq
rt[5 + 4*Sqrt[1 - 9*y[x]] - 9*y[x]]*Sqrt[x^2 + 3*y[x]])])/Sqrt[5 + 4*Sqrt[1 - 9*
y[x]] - 9*y[x]])*y[x])/(2*Sqrt[1 - 9*y[x]]) == 0, y[x]]}

Maple raw input

dsolve(3*diff(y(x),x)^2+4*x*diff(y(x),x)+x^2-y(x) = 0, y(x))

Maple raw output

[y(x) = -1/3*x^2, y(x) = -1/12*(5*_C1^2*x^2+2*_C1*x*(-x*_C1+3^(1/2))-3)/_C1^2, y
(x) = -1/12*(5*_C1^2*x^2-2*_C1*x*(x*_C1+3^(1/2))-3)/_C1^2, y(x) = -5/12*x^2+1/6*
x*(x-3^(1/2)*_C1)+1/4*_C1^2, y(x) = -5/12*x^2+1/6*x*(x+3^(1/2)*_C1)+1/4*_C1^2]