4.15.9 $(x^2-y(x{)}^4)y^{\prime }(x)=xy(x)$

ODE
$$(x^2-y(x{)}^4)y^{\prime }(x)=xy(x)$$ ODE Classification

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

Book solution method
Change of Variable, new independent variable

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

$$\{\{y(x)\to -\sqrt{-\sqrt{c_1^2-x^2}-c_1}\},\{y(x)\to \sqrt{-\sqrt{c_1^2-x^2}-c_1}\},\{y(x)\to -\sqrt{\sqrt{c_1^2-x^2}-c_1}\},\{y(x)\to \sqrt{\sqrt{c_1^2-x^2}-c_1}\}\}$$

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

$$\{\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}+\ln \left(\frac{{\left(y\left(x\right)\right)}^4+x^2}{x^2}\right)-2\ln \left(\frac{y\left(x\right)}{\sqrt{x}}\right)=0\}$$ 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