##### 4.19.10 $$x^3+x^2 y'(x)^2-3 x y(x) y'(x)+2 y(x)^2=0$$

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

[[_homogeneous, class G], _rational]

Book solution method
Change of variable

Mathematica
cpu = 1.06072 (sec), leaf count = 351

$\left \{\text {Solve}\left [c_1=\int _1^{y(x)}\frac {\left (x K[3]^2-4 x^4\right ) \int _1^x\frac {12 K[2] \sqrt {K[2]^2 K[3]^2-4 K[2]^5}}{\left (K[3]^2-4 K[2]^3\right )^2}dK[2]-2 \sqrt {x^2 K[3]^2-4 x^5}}{4 x^4-x K[3]^2}dK[3]+\int _1^x\frac {4 K[2]^4-y(x)^2 K[2]+3 y(x) \sqrt {K[2]^2 \left (y(x)^2-4 K[2]^3\right )}}{4 K[2]^5-K[2]^2 y(x)^2}dK[2],y(x)\right ],\text {Solve}\left [c_1=\int _1^{y(x)}\frac {\left (x K[5]^2-4 x^4\right ) \int _1^x-\frac {12 K[4] \sqrt {K[4]^2 K[5]^2-4 K[4]^5}}{\left (K[5]^2-4 K[4]^3\right )^2}dK[4]+2 \sqrt {x^2 K[5]^2-4 x^5}}{4 x^4-x K[5]^2}dK[5]+\int _1^x\frac {4 K[4]^4-y(x)^2 K[4]-3 y(x) \sqrt {K[4]^2 \left (y(x)^2-4 K[4]^3\right )}}{4 K[4]^5-K[4]^2 y(x)^2}dK[4],y(x)\right ]\right \}$

Maple
cpu = 1.689 (sec), leaf count = 49

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

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

Mathematica raw output

{Solve[C[1] == Inactive[Integrate][(4*K[2]^4 - K[2]*y[x]^2 + 3*y[x]*Sqrt[K[2]^2*
(-4*K[2]^3 + y[x]^2)])/(4*K[2]^5 - K[2]^2*y[x]^2), {K[2], 1, x}] + Inactive[Inte
grate][(-2*Sqrt[-4*x^5 + x^2*K[3]^2] + (-4*x^4 + x*K[3]^2)*Inactive[Integrate][(
12*K[2]*Sqrt[-4*K[2]^5 + K[2]^2*K[3]^2])/(-4*K[2]^3 + K[3]^2)^2, {K[2], 1, x}])/
(4*x^4 - x*K[3]^2), {K[3], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(
4*K[4]^4 - K[4]*y[x]^2 - 3*y[x]*Sqrt[K[4]^2*(-4*K[4]^3 + y[x]^2)])/(4*K[4]^5 - K
[4]^2*y[x]^2), {K[4], 1, x}] + Inactive[Integrate][(2*Sqrt[-4*x^5 + x^2*K[5]^2]
+ (-4*x^4 + x*K[5]^2)*Inactive[Integrate][(-12*K[4]*Sqrt[-4*K[4]^5 + K[4]^2*K[5]
^2])/(-4*K[4]^3 + K[5]^2)^2, {K[4], 1, x}])/(4*x^4 - x*K[5]^2), {K[5], 1, y[x]}]
, y[x]]}

Maple raw input

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

Maple raw output

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