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 = 0.0892305 (sec), leaf count = 61

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

Maple
cpu = 0.012 (sec), leaf count = 33

\[ \left \{ -\ln \left ({\frac {{x}^{2}+ \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}} \right ) +\ln \left ({\frac {y \relax (x ) }{x}} \right ) -\ln \relax (x ) -{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

-ln((x^2+y(x)^2)/x^2)+ln(y(x)/x)-ln(x)-_C1 = 0