4.14.4 $2x(5x^2+y(x{)}^2)y^{\prime }(x)=x^{2}y(x)-y(x{)}^3$

ODE
$$2x(5x^2+y(x{)}^2)y^{\prime }(x)=x^{2}y(x)-y(x{)}^3$$ ODE Classification

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Homogeneous equation

Mathematica ✓
cpu = 0.0608681 (sec), leaf count = 216

$$\{\{y(x)\to \text{Root}[-{\mathrm{\#}\text{1}}^5+\frac{{\mathrm{\#}\text{1}}^2{e}^{3c_1}}{x^{3/2}}+3e^{3c_1}\sqrt{x}\mathrm{\&},1]\},\{y(x)\to \text{Root}[-{\mathrm{\#}\text{1}}^5+\frac{{\mathrm{\#}\text{1}}^2{e}^{3c_1}}{x^{3/2}}+3e^{3c_1}\sqrt{x}\mathrm{\&},2]\},\{y(x)\to \text{Root}[-{\mathrm{\#}\text{1}}^5+\frac{{\mathrm{\#}\text{1}}^2{e}^{3c_1}}{x^{3/2}}+3e^{3c_1}\sqrt{x}\mathrm{\&},3]\},\{y(x)\to \text{Root}[-{\mathrm{\#}\text{1}}^5+\frac{{\mathrm{\#}\text{1}}^2{e}^{3c_1}}{x^{3/2}}+3e^{3c_1}\sqrt{x}\mathrm{\&},4]\},\{y(x)\to \text{Root}[-{\mathrm{\#}\text{1}}^5+\frac{{\mathrm{\#}\text{1}}^2{e}^{3c_1}}{x^{3/2}}+3e^{3c_1}\sqrt{x}\mathrm{\&},5]\}\}$$

Maple ✓
cpu = 0.021 (sec), leaf count = 37

$$\{\frac{2}{9}\ln \left(\frac{3x^2+{\left(y\left(x\right)\right)}^2}{x^2}\right)-\frac{10}{9}\ln \left(\frac{y\left(x\right)}{x}\right)-\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[2*x*(5*x^2 + y[x]^2)*y'[x] == x^2*y[x] - y[x]^3,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(2*x*(5*x^2+y(x)^2)*diff(y(x),x) = x^2*y(x)-y(x)^3, y(x),'implicit')

Maple raw output

2/9*ln((3*x^2+y(x)^2)/x^2)-10/9*ln(y(x)/x)-ln(x)-_C1 = 0