##### 4.18.11 $$x^3+x y'(x)^2+y(x) y'(x)=0$$

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

[[_homogeneous, class G]]

Book solution method
Homogeneous ODE, The Isobaric equation

Mathematica
cpu = 0.296827 (sec), leaf count = 107

$\left \{\left \{y(x)\to x^2 \text {InverseFunction}\left [\int _1^{\text {\#1}}\frac {1}{5 K[2]+\sqrt {K[2]^2-4}}dK[2]\& \right ]\left [\int _1^x-\frac {1}{2 K[3]}dK[3]+c_1\right ]\right \},\left \{y(x)\to x^2 \text {InverseFunction}\left [\int _1^{\text {\#1}}\frac {1}{\sqrt {K[4]^2-4}-5 K[4]}dK[4]\& \right ]\left [\int _1^x\frac {1}{2 K[5]}dK[5]+c_1\right ]\right \}\right \}$

Maple
cpu = 0.299 (sec), leaf count = 337

$\left [\int _{\textit {\_b}}^{x}-\frac {-y \left (x \right )+\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}}{\textit {\_a} \left (-5 y \left (x \right )+\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}\right )}d \textit {\_a} +\int _{}^{y \left (x \right )}-\frac {2 \left (8 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (-5 \textit {\_f} +\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}\right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \sqrt {-4 x^{4}+\textit {\_f}^{2}}-40 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (-5 \textit {\_f} +\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}\right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \textit {\_f} -1\right )}{-5 \textit {\_f} +\sqrt {-4 x^{4}+\textit {\_f}^{2}}}d \textit {\_f} +\textit {\_C1} = 0, \int _{\textit {\_b}}^{x}-\frac {y \left (x \right )+\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}}{\left (\sqrt {-4 \textit {\_a}^{4}+y \left (x \right )^{2}}+5 y \left (x \right )\right ) \textit {\_a}}d \textit {\_a} +\int _{}^{y \left (x \right )}\frac {16 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}+5 \textit {\_f} \right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \sqrt {-4 x^{4}+\textit {\_f}^{2}}+80 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}+5 \textit {\_f} \right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \textit {\_f} -2}{\sqrt {-4 x^{4}+\textit {\_f}^{2}}+5 \textit {\_f}}d \textit {\_f} +\textit {\_C1} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> x^2*InverseFunction[Inactive[Integrate][(5*K[2] + Sqrt[-4 + K[2]^2])^(
-1), {K[2], 1, #1}] & ][C[1] + Inactive[Integrate][-1/2*1/K[3], {K[3], 1, x}]]},
 {y[x] -> x^2*InverseFunction[Inactive[Integrate][(-5*K[4] + Sqrt[-4 + K[4]^2])^
(-1), {K[4], 1, #1}] & ][C[1] + Inactive[Integrate][1/(2*K[5]), {K[5], 1, x}]]}}

Maple raw input

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

Maple raw output

[Int(-(-y(x)+(-4*_a^4+y(x)^2)^(1/2))/_a/(-5*y(x)+(-4*_a^4+y(x)^2)^(1/2)),_a = _b
 .. x)+Intat(-2*(8*Int(_a^3/(-5*_f+(-4*_a^4+_f^2)^(1/2))^2/(-4*_a^4+_f^2)^(1/2),
_a = _b .. x)*(-4*x^4+_f^2)^(1/2)-40*Int(_a^3/(-5*_f+(-4*_a^4+_f^2)^(1/2))^2/(-4
*_a^4+_f^2)^(1/2),_a = _b .. x)*_f-1)/(-5*_f+(-4*x^4+_f^2)^(1/2)),_f = y(x))+_C1
 = 0, Int(-1/((-4*_a^4+y(x)^2)^(1/2)+5*y(x))*(y(x)+(-4*_a^4+y(x)^2)^(1/2))/_a,_a
 = _b .. x)+Intat(2*(8*Int(1/((-4*_a^4+_f^2)^(1/2)+5*_f)^2*_a^3/(-4*_a^4+_f^2)^(
1/2),_a = _b .. x)*(-4*x^4+_f^2)^(1/2)+40*Int(1/((-4*_a^4+_f^2)^(1/2)+5*_f)^2*_a
^3/(-4*_a^4+_f^2)^(1/2),_a = _b .. x)*_f-1)/((-4*x^4+_f^2)^(1/2)+5*_f),_f = y(x)
)+_C1 = 0]