4.9.10 \(\left (x^3+1\right )^{2/3} y'(x)+\left (y(x)^3+1\right )^{2/3}=0\)

ODE
\[ \left (x^3+1\right )^{2/3} y'(x)+\left (y(x)^3+1\right )^{2/3}=0 \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.782776 (sec), leaf count = 196

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {3 \sqrt [3]{\frac {\sqrt [3]{-1}-\text {$\#$1}}{1+\sqrt [3]{-1}}} (\text {$\#$1}+1) \left (\frac {\text {$\#$1}+(-1)^{2/3}}{(-1)^{2/3}-1}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {\sqrt [3]{-1} (\text {$\#$1}+1)}{\left (-1+\sqrt [3]{-1}\right ) \text {$\#$1}+1}\right )}{\left (\text {$\#$1}^3+1\right )^{2/3}}\& \right ]\left [c_1-\frac {3 \sqrt [3]{\frac {\sqrt [3]{-1}-x}{1+\sqrt [3]{-1}}} (x+1) \left (\frac {x+(-1)^{2/3}}{(-1)^{2/3}-1}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {\sqrt [3]{-1} (x+1)}{\left (-1+\sqrt [3]{-1}\right ) x+1}\right )}{\left (x^3+1\right )^{2/3}}\right ]\right \}\right \}\]

Maple
cpu = 0.062 (sec), leaf count = 33

\[ \left \{ x{\mbox {$_2$F$_1$}({\frac {1}{3}},{\frac {2}{3}};\,{\frac {4}{3}};\,-{x}^{3})}+y \relax (x ) {\mbox {$_2$F$_1$}({\frac {1}{3}},{\frac {2}{3}};\,{\frac {4}{3}};\,- \left (y \relax (x ) \right ) ^{3})}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[(1 + y[x]^3)^(2/3) + (1 + x^3)^(2/3)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[(3*Hypergeometric2F1[1/3, 2/3, 4/3, ((-1)^(1/3)*(1 + #
1))/(1 + (-1 + (-1)^(1/3))*#1)]*(((-1)^(1/3) - #1)/(1 + (-1)^(1/3)))^(1/3)*(1 + 
#1)*(((-1)^(2/3) + #1)/(-1 + (-1)^(2/3)))^(2/3))/(1 + #1^3)^(2/3) & ][C[1] - (3*
(((-1)^(1/3) - x)/(1 + (-1)^(1/3)))^(1/3)*(1 + x)*(((-1)^(2/3) + x)/(-1 + (-1)^(
2/3)))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, ((-1)^(1/3)*(1 + x))/(1 + (-1 + (-
1)^(1/3))*x)])/(1 + x^3)^(2/3)]}}

Maple raw input

dsolve(diff(y(x),x)*(x^3+1)^(2/3)+(1+y(x)^3)^(2/3) = 0, y(x),'implicit')

Maple raw output

x*hypergeom([1/3, 2/3],[4/3],-x^3)+y(x)*hypergeom([1/3, 2/3],[4/3],-y(x)^3)+_C1 
= 0