4.14.4 \(2 x \left (5 x^2+y(x)^2\right ) y'(x)=x^2 y(x)-y(x)^3\)

ODE
\[ 2 x \left (5 x^2+y(x)^2\right ) y'(x)=x^2 y(x)-y(x)^3 \] ODE Classification

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

Book solution method
Homogeneous equation

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

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

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

\[ \left \{ {\frac {2}{9}\ln \left ({\frac {3\,{x}^{2}+ \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}} \right ) }-{\frac {10}{9}\ln \left ({\frac {y \relax (x ) }{x}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] 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