##### 4.15.11 $$\left (a^2 x^2+\left (x^2+y(x)^2\right )^2\right ) y'(x)=a^2 x y(x)$$

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

[_rational]

Book solution method
Change of Variable, Two new variables

Mathematica
cpu = 0.596791 (sec), leaf count = 239

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

Maple
cpu = 0.642 (sec), leaf count = 197

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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