4.15.18 $(x^2-y(x{)}^5)y^{\prime }(x)=2xy(x)$

ODE
$$(x^2-y(x{)}^5)y^{\prime }(x)=2xy(x)$$ ODE Classification

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

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 0.0127628 (sec), leaf count = 116

$$\{\{y(x)\to \text{Root}[{\mathrm{\#}\text{1}}^5+4\mathrm{\#}\text{1}{c}_1+4x^2\mathrm{\&},1]\},\{y(x)\to \text{Root}[{\mathrm{\#}\text{1}}^5+4\mathrm{\#}\text{1}{c}_1+4x^2\mathrm{\&},2]\},\{y(x)\to \text{Root}[{\mathrm{\#}\text{1}}^5+4\mathrm{\#}\text{1}{c}_1+4x^2\mathrm{\&},3]\},\{y(x)\to \text{Root}[{\mathrm{\#}\text{1}}^5+4\mathrm{\#}\text{1}{c}_1+4x^2\mathrm{\&},4]\},\{y(x)\to \text{Root}[{\mathrm{\#}\text{1}}^5+4\mathrm{\#}\text{1}{c}_1+4x^2\mathrm{\&},5]\}\}$$

Maple ✓
cpu = 0.02 (sec), leaf count = 35

$$\{\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}-\frac{5}{8}\ln \left(y\left(x\right)x^{-\frac{2}{5}}\right)+\frac{5}{8}\ln \left(\frac{{\left(y\left(x\right)\right)}^5+4x^2}{x^2}\right)=0\}$$ Mathematica raw input

DSolve[(x^2 - y[x]^5)*y'[x] == 2*x*y[x],y[x],x]

Mathematica raw output

{{y[x] -> Root[4*x^2 + 4*C[1]*#1 + #1^5 & , 1]}, {y[x] -> Root[4*x^2 + 4*C[1]*#1
 + #1^5 & , 2]}, {y[x] -> Root[4*x^2 + 4*C[1]*#1 + #1^5 & , 3]}, {y[x] -> Root[4
*x^2 + 4*C[1]*#1 + #1^5 & , 4]}, {y[x] -> Root[4*x^2 + 4*C[1]*#1 + #1^5 & , 5]}}

Maple raw input

dsolve((x^2-y(x)^5)*diff(y(x),x) = 2*x*y(x), y(x),'implicit')

Maple raw output

ln(x)-_C1-5/8*ln(y(x)/x^(2/5))+5/8*ln((y(x)^5+4*x^2)/x^2) = 0