4.15.20 \(y(x)^3 \left (3 x^5 y(x)^5-1\right )+x^3 \left (5 x^3 y(x)^7+1\right ) y'(x)=0\)

ODE
\[ y(x)^3 \left (3 x^5 y(x)^5-1\right )+x^3 \left (5 x^3 y(x)^7+1\right ) y'(x)=0 \] ODE Classification

[_rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0246112 (sec), leaf count = 253

\[\left \{\left \{y(x)\to \text {Root}\left [2 \text {$\#$1}^7 x^5+\text {$\#$1}^2 \left (1-2 c_1 x^2\right )-x^2\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [2 \text {$\#$1}^7 x^5+\text {$\#$1}^2 \left (1-2 c_1 x^2\right )-x^2\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [2 \text {$\#$1}^7 x^5+\text {$\#$1}^2 \left (1-2 c_1 x^2\right )-x^2\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [2 \text {$\#$1}^7 x^5+\text {$\#$1}^2 \left (1-2 c_1 x^2\right )-x^2\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [2 \text {$\#$1}^7 x^5+\text {$\#$1}^2 \left (1-2 c_1 x^2\right )-x^2\& ,5\right ]\right \},\left \{y(x)\to \text {Root}\left [2 \text {$\#$1}^7 x^5+\text {$\#$1}^2 \left (1-2 c_1 x^2\right )-x^2\& ,6\right ]\right \},\left \{y(x)\to \text {Root}\left [2 \text {$\#$1}^7 x^5+\text {$\#$1}^2 \left (1-2 c_1 x^2\right )-x^2\& ,7\right ]\right \}\right \}\]

Maple
cpu = 0.172 (sec), leaf count = 25

\[ \left \{ -{x}^{3} \left (y \relax (x ) \right ) ^{5}-{\frac {1}{2\,{x}^{2}}}+{\frac {1}{2\, \left (y \relax (x ) \right ) ^{2}}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[y[x]^3*(-1 + 3*x^5*y[x]^5) + x^3*(1 + 5*x^3*y[x]^7)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> Root[-x^2 + (1 - 2*x^2*C[1])*#1^2 + 2*x^5*#1^7 & , 1]}, {y[x] -> Root[
-x^2 + (1 - 2*x^2*C[1])*#1^2 + 2*x^5*#1^7 & , 2]}, {y[x] -> Root[-x^2 + (1 - 2*x
^2*C[1])*#1^2 + 2*x^5*#1^7 & , 3]}, {y[x] -> Root[-x^2 + (1 - 2*x^2*C[1])*#1^2 +
 2*x^5*#1^7 & , 4]}, {y[x] -> Root[-x^2 + (1 - 2*x^2*C[1])*#1^2 + 2*x^5*#1^7 & ,
 5]}, {y[x] -> Root[-x^2 + (1 - 2*x^2*C[1])*#1^2 + 2*x^5*#1^7 & , 6]}, {y[x] -> 
Root[-x^2 + (1 - 2*x^2*C[1])*#1^2 + 2*x^5*#1^7 & , 7]}}

Maple raw input

dsolve(x^3*(1+5*x^3*y(x)^7)*diff(y(x),x)+(3*x^5*y(x)^5-1)*y(x)^3 = 0, y(x),'implicit')

Maple raw output

-x^3*y(x)^5-1/2/x^2+1/2/y(x)^2+_C1 = 0