4.15.9 \(\left (x^2-y(x)^4\right ) y'(x)=x y(x)\)

ODE
\[ \left (x^2-y(x)^4\right ) y'(x)=x y(x) \] ODE Classification

[[_homogeneous, `class G`], _rational]

Book solution method
Change of Variable, new independent variable

Mathematica
cpu = 0.639934 (sec), leaf count = 117

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

Maple
cpu = 0.017 (sec), leaf count = 31

\[ \left \{ \ln \relax (x ) -{\it \_C1}+\ln \left ({\frac { \left (y \relax (x ) \right ) ^{4}+{x}^{2}}{{x}^{2}}} \right ) -2\,\ln \left ({\frac {y \relax (x ) }{\sqrt {x}}} \right ) =0 \right \} \] Mathematica raw input

DSolve[(x^2 - y[x]^4)*y'[x] == x*y[x],y[x],x]

Mathematica raw output

{{y[x] -> -Sqrt[-C[1] - Sqrt[-x^2 + C[1]^2]]}, {y[x] -> Sqrt[-C[1] - Sqrt[-x^2 +
 C[1]^2]]}, {y[x] -> -Sqrt[-C[1] + Sqrt[-x^2 + C[1]^2]]}, {y[x] -> Sqrt[-C[1] + 
Sqrt[-x^2 + C[1]^2]]}}

Maple raw input

dsolve((x^2-y(x)^4)*diff(y(x),x) = x*y(x), y(x),'implicit')

Maple raw output

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