4.12.45 \(\sqrt {x^2+1} (y(x)+1) y'(x)=y(x)^3\)

ODE
\[ \sqrt {x^2+1} (y(x)+1) y'(x)=y(x)^3 \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.0374284 (sec), leaf count = 63

\[\left \{\left \{y(x)\to -\frac {\sqrt {-2 c_1-2 \sinh ^{-1}(x)+1}+1}{2 \left (c_1+\sinh ^{-1}(x)\right )}\right \},\left \{y(x)\to \frac {\sqrt {-2 c_1-2 \sinh ^{-1}(x)+1}-1}{2 \left (c_1+\sinh ^{-1}(x)\right )}\right \}\right \}\]

Maple
cpu = 0.013 (sec), leaf count = 17

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

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

Mathematica raw output

{{y[x] -> -(1 + Sqrt[1 - 2*ArcSinh[x] - 2*C[1]])/(2*(ArcSinh[x] + C[1]))}, {y[x]
 -> (-1 + Sqrt[1 - 2*ArcSinh[x] - 2*C[1]])/(2*(ArcSinh[x] + C[1]))}}

Maple raw input

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

Maple raw output

arcsinh(x)+1/y(x)+1/2/y(x)^2+_C1 = 0