4.9.3 \(\sqrt {x^3+1} y'(x)=\sqrt {y(x)^3+1}\)

ODE
\[ \sqrt {x^3+1} y'(x)=\sqrt {y(x)^3+1} \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.334943 (sec), leaf count = 191

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {2 \sqrt [6]{-1} \sqrt {-\sqrt [6]{-1} \left (\text {$\#$1}+(-1)^{2/3}\right )} \sqrt {(-1)^{2/3} \text {$\#$1}^2+\sqrt [3]{-1} \text {$\#$1}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {-(-1)^{5/6} (\text {$\#$1}+1)}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3} \sqrt {\text {$\#$1}^3+1}}\& \right ]\left [c_1+\frac {2 \sqrt [6]{-1} \sqrt {-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )} \sqrt {(-1)^{2/3} x^2+\sqrt [3]{-1} x+1} F\left (\sin ^{-1}\left (\frac {\sqrt {-(-1)^{5/6} (x+1)}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3} \sqrt {x^3+1}}\right ]\right \}\right \}\]

Maple
cpu = 0.013 (sec), leaf count = 28

\[ \left \{ \int \!{\frac {1}{\sqrt {{x}^{3}+1}}}\,{\rm d}x-\int ^{y \relax (x ) }\!{\frac {1}{\sqrt {{{\it \_a}}^{3}+1}}}{d{\it \_a}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[Sqrt[1 + x^3]*y'[x] == Sqrt[1 + y[x]^3],y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[(2*(-1)^(1/6)*EllipticF[ArcSin[Sqrt[-((-1)^(5/6)*(1 + 
#1))]/3^(1/4)], (-1)^(1/3)]*Sqrt[-((-1)^(1/6)*((-1)^(2/3) + #1))]*Sqrt[1 + (-1)^
(1/3)*#1 + (-1)^(2/3)*#1^2])/(3^(1/4)*Sqrt[1 + #1^3]) & ][C[1] + (2*(-1)^(1/6)*S
qrt[-((-1)^(1/6)*((-1)^(2/3) + x))]*Sqrt[1 + (-1)^(1/3)*x + (-1)^(2/3)*x^2]*Elli
pticF[ArcSin[Sqrt[-((-1)^(5/6)*(1 + x))]/3^(1/4)], (-1)^(1/3)])/(3^(1/4)*Sqrt[1 
+ x^3])]}}

Maple raw input

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

Maple raw output

Int(1/(x^3+1)^(1/2),x)-Intat(1/(_a^3+1)^(1/2),_a = y(x))+_C1 = 0