##### 4.15.9 $$\left (x^2-y(x)^4\right ) y'(x)=x y(x)$$

ODE
$\left (x^2-y(x)^4\right ) y'(x)=x y(x)$ ODE Classiﬁcation

[[_homogeneous, class G], _rational]

Book solution method
Change of Variable, new independent variable

Mathematica
cpu = 0.315953 (sec), leaf count = 117

$\left \{\left \{y(x)\to -\sqrt {-\sqrt {-x^2+c_1{}^2}-c_1}\right \},\left \{y(x)\to \sqrt {-\sqrt {-x^2+c_1{}^2}-c_1}\right \},\left \{y(x)\to -\sqrt {\sqrt {-x^2+c_1{}^2}-c_1}\right \},\left \{y(x)\to \sqrt {\sqrt {-x^2+c_1{}^2}-c_1}\right \}\right \}$

Maple
cpu = 0.093 (sec), leaf count = 97

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

DSolve[(x^2 - y[x]^4)*y'[x] == x*y[x],y[x],x]

Mathematica raw output

{{y[x] -> -Sqrt[-C[1] - Sqrt[-x^2 + C[1]^2]]}, {y[x] -> Sqrt[-C[1] - Sqrt[-x^2 +
C[1]^2]]}, {y[x] -> -Sqrt[-C[1] + Sqrt[-x^2 + C[1]^2]]}, {y[x] -> Sqrt[-C[1] +
Sqrt[-x^2 + C[1]^2]]}}

Maple raw input

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

Maple raw output

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