##### 4.14.33 $$y(x) \left (a+x^2+y(x)^2\right ) y'(x)=x \left (a-x^2-y(x)^2\right )$$

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

[_exact, _rational]

Book solution method
Exact equation

Mathematica
cpu = 0.523207 (sec), leaf count = 149

$\left \{\left \{y(x)\to -\sqrt {-\sqrt {a^2+4 a x^2+4 c_1}-a-x^2}\right \},\left \{y(x)\to \sqrt {-\sqrt {a^2+4 a x^2+4 c_1}-a-x^2}\right \},\left \{y(x)\to -\sqrt {\sqrt {a^2+4 a x^2+4 c_1}-a-x^2}\right \},\left \{y(x)\to \sqrt {\sqrt {a^2+4 a x^2+4 c_1}-a-x^2}\right \}\right \}$

Maple
cpu = 0.055 (sec), leaf count = 125

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

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

Mathematica raw output

{{y[x] -> -Sqrt[-a - x^2 - Sqrt[a^2 + 4*a*x^2 + 4*C[1]]]}, {y[x] -> Sqrt[-a - x^
2 - Sqrt[a^2 + 4*a*x^2 + 4*C[1]]]}, {y[x] -> -Sqrt[-a - x^2 + Sqrt[a^2 + 4*a*x^2
 + 4*C[1]]]}, {y[x] -> Sqrt[-a - x^2 + Sqrt[a^2 + 4*a*x^2 + 4*C[1]]]}}

Maple raw input

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

Maple raw output

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