##### 4.18.10 $$-x^2+x y'(x)^2+y(x) y'(x)=0$$

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

[[_homogeneous, class G]]

Book solution method
Homogeneous ODE, The Isobaric equation

Mathematica
cpu = 0.28221 (sec), leaf count = 479

$\left \{\text {Solve}\left [c_1=\int \frac {4 x^3-3 y(x)^2+3 y(x) \sqrt {4 x^3+y(x)^2}}{4 x^4-15 x y(x)^2}dx+\int \frac {4 \int \frac {12 x^2 \left (4 x^3+17 y(x)^2+8 y(x) \sqrt {4 x^3+y(x)^2}\right )}{\left (4 x^3-15 y(x)^2\right )^2 \sqrt {4 x^3+y(x)^2}}dx x^3+8 y(x)-15 y(x)^2 \int \frac {12 x^2 \left (4 x^3+17 y(x)^2+8 y(x) \sqrt {4 x^3+y(x)^2}\right )}{\left (4 x^3-15 y(x)^2\right )^2 \sqrt {4 x^3+y(x)^2}}dx+2 \sqrt {4 x^3+y(x)^2}}{15 y(x)^2-4 x^3}dy(x),y(x)\right ],\text {Solve}\left [c_1=\int \frac {4 x^3-3 y(x)^2-3 y(x) \sqrt {4 x^3+y(x)^2}}{4 x^4-15 x y(x)^2}dx+\int \frac {-4 \int \frac {12 x^2 \left (-4 x^3-17 y(x)^2+8 y(x) \sqrt {4 x^3+y(x)^2}\right )}{\left (4 x^3-15 y(x)^2\right )^2 \sqrt {4 x^3+y(x)^2}}dx x^3-8 y(x)+15 y(x)^2 \int \frac {12 x^2 \left (-4 x^3-17 y(x)^2+8 y(x) \sqrt {4 x^3+y(x)^2}\right )}{\left (4 x^3-15 y(x)^2\right )^2 \sqrt {4 x^3+y(x)^2}}dx+2 \sqrt {4 x^3+y(x)^2}}{4 x^3-15 y(x)^2}dy(x),y(x)\right ]\right \}$

Maple
cpu = 0.267 (sec), leaf count = 337

$\left [\int _{\textit {\_b}}^{x}-\frac {-y \left (x \right )+\sqrt {4 \textit {\_a}^{3}+y \left (x \right )^{2}}}{\textit {\_a} \left (-4 y \left (x \right )+\sqrt {4 \textit {\_a}^{3}+y \left (x \right )^{2}}\right )}d \textit {\_a} +\int _{}^{y \left (x \right )}\frac {12 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{2}}{\left (-4 \textit {\_f} +\sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}\right )^{2} \sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \sqrt {4 x^{3}+\textit {\_f}^{2}}-48 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{2}}{\left (-4 \textit {\_f} +\sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}\right )^{2} \sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \textit {\_f} +2}{-4 \textit {\_f} +\sqrt {4 x^{3}+\textit {\_f}^{2}}}d \textit {\_f} +\textit {\_C1} = 0, \int _{\textit {\_b}}^{x}-\frac {y \left (x \right )+\sqrt {4 \textit {\_a}^{3}+y \left (x \right )^{2}}}{\left (\sqrt {4 \textit {\_a}^{3}+y \left (x \right )^{2}}+4 y \left (x \right )\right ) \textit {\_a}}d \textit {\_a} +\int _{}^{y \left (x \right )}-\frac {2 \left (6 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{2}}{\left (\sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}+4 \textit {\_f} \right )^{2} \sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \sqrt {4 x^{3}+\textit {\_f}^{2}}+24 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{2}}{\left (\sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}+4 \textit {\_f} \right )^{2} \sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \textit {\_f} +1\right )}{\sqrt {4 x^{3}+\textit {\_f}^{2}}+4 \textit {\_f}}d \textit {\_f} +\textit {\_C1} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

{Solve[C[1] == Inactive[Integrate][(4*x^3 - 3*y[x]^2 + 3*y[x]*Sqrt[4*x^3 + y[x]^
2])/(4*x^4 - 15*x*y[x]^2), x] + Inactive[Integrate][(8*y[x] + 2*Sqrt[4*x^3 + y[x
]^2] + 4*x^3*Inactive[Integrate][(12*x^2*(4*x^3 + 17*y[x]^2 + 8*y[x]*Sqrt[4*x^3
+ y[x]^2]))/((4*x^3 - 15*y[x]^2)^2*Sqrt[4*x^3 + y[x]^2]), x] - 15*y[x]^2*Inactiv
e[Integrate][(12*x^2*(4*x^3 + 17*y[x]^2 + 8*y[x]*Sqrt[4*x^3 + y[x]^2]))/((4*x^3
- 15*y[x]^2)^2*Sqrt[4*x^3 + y[x]^2]), x])/(-4*x^3 + 15*y[x]^2), y[x]], y[x]], So
lve[C[1] == Inactive[Integrate][(4*x^3 - 3*y[x]^2 - 3*y[x]*Sqrt[4*x^3 + y[x]^2])
/(4*x^4 - 15*x*y[x]^2), x] + Inactive[Integrate][(-8*y[x] + 2*Sqrt[4*x^3 + y[x]^
2] - 4*x^3*Inactive[Integrate][(12*x^2*(-4*x^3 - 17*y[x]^2 + 8*y[x]*Sqrt[4*x^3 +
 y[x]^2]))/((4*x^3 - 15*y[x]^2)^2*Sqrt[4*x^3 + y[x]^2]), x] + 15*y[x]^2*Inactive
[Integrate][(12*x^2*(-4*x^3 - 17*y[x]^2 + 8*y[x]*Sqrt[4*x^3 + y[x]^2]))/((4*x^3
- 15*y[x]^2)^2*Sqrt[4*x^3 + y[x]^2]), x])/(4*x^3 - 15*y[x]^2), y[x]], y[x]]}

Maple raw input

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

Maple raw output

[Int(-(-y(x)+(4*_a^3+y(x)^2)^(1/2))/_a/(-4*y(x)+(4*_a^3+y(x)^2)^(1/2)),_a = _b .
. x)+Intat(2*(6*Int(_a^2/(-4*_f+(4*_a^3+_f^2)^(1/2))^2/(4*_a^3+_f^2)^(1/2),_a =
_b .. x)*(4*x^3+_f^2)^(1/2)-24*Int(_a^2/(-4*_f+(4*_a^3+_f^2)^(1/2))^2/(4*_a^3+_f
^2)^(1/2),_a = _b .. x)*_f+1)/(-4*_f+(4*x^3+_f^2)^(1/2)),_f = y(x))+_C1 = 0, Int
(-1/((4*_a^3+y(x)^2)^(1/2)+4*y(x))*(y(x)+(4*_a^3+y(x)^2)^(1/2))/_a,_a = _b .. x)
+Intat(-2*(6*Int(1/((4*_a^3+_f^2)^(1/2)+4*_f)^2*_a^2/(4*_a^3+_f^2)^(1/2),_a = _b
 .. x)*(4*x^3+_f^2)^(1/2)+24*Int(1/((4*_a^3+_f^2)^(1/2)+4*_f)^2*_a^2/(4*_a^3+_f^
2)^(1/2),_a = _b .. x)*_f+1)/((4*x^3+_f^2)^(1/2)+4*_f),_f = y(x))+_C1 = 0]