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 = 0.477636 (sec), leaf count = 121

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

Maple
cpu = 0.02 (sec), leaf count = 37

\[ \left \{ -{\frac {1}{4}\ln \left ({\frac {2\,{x}^{2}+ \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}} \right ) }-{\frac {1}{2}\ln \left ({\frac {y \relax (x ) }{x}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

-1/4*ln((2*x^2+y(x)^2)/x^2)-1/2*ln(y(x)/x)-ln(x)-_C1 = 0