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

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

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

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0161518 (sec), leaf count = 141

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

Maple
cpu = 0.02 (sec), leaf count = 37

\[ \left \{ \ln \relax (x ) -{\it \_C1}-{\frac {5}{2}\ln \left ({y \relax (x ) {x}^{-{\frac {3}{5}}}} \right ) }+{\frac {5}{8}\ln \left ({\frac {4\, \left (y \relax (x ) \right ) ^{5}-{x}^{3}}{{x}^{3}}} \right ) }=0 \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

ln(x)-_C1-5/2*ln(y(x)/x^(3/5))+5/8*ln((4*y(x)^5-x^3)/x^3) = 0