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