##### 4.13.3 $$\left (x^2-y(x)^2\right ) y'(x)=2 x y(x)$$

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

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.347074 (sec), leaf count = 61

$\left \{\left \{y(x)\to \frac {1}{2} \left (e^{c_1}-\sqrt {-4 x^2+e^{2 c_1}}\right )\right \},\left \{y(x)\to \frac {1}{2} \left (\sqrt {-4 x^2+e^{2 c_1}}+e^{c_1}\right )\right \}\right \}$

Maple
cpu = 0.044 (sec), leaf count = 45

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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