##### 4.13.1 $$\left (x^2+y(x)^2\right ) y'(x)+x y(x)=0$$

ODE
$\left (x^2+y(x)^2\right ) y'(x)+x y(x)=0$ ODE Classiﬁcation

[[_homogeneous, class A], _rational, _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.397409 (sec), leaf count = 121

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

Maple
cpu = 0.282 (sec), leaf count = 223

$\left [y \left (x \right ) = \frac {\sqrt {\left (x^{2} \textit {\_C1} -\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1} \,x^{2}}}{x \left (x^{2} \textit {\_C1} -\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1}}, y \left (x \right ) = \frac {\sqrt {\left (x^{2} \textit {\_C1} +\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1} \,x^{2}}}{x \left (x^{2} \textit {\_C1} +\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1}}, y \left (x \right ) = -\frac {\sqrt {\left (x^{2} \textit {\_C1} -\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1} \,x^{2}}}{x \left (x^{2} \textit {\_C1} -\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1}}, y \left (x \right ) = -\frac {\sqrt {\left (x^{2} \textit {\_C1} +\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1} \,x^{2}}}{x \left (x^{2} \textit {\_C1} +\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1}}\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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