4.16.1 \((\sinh (x)+1) y'(x) \sinh (y(x))+\cosh (x) (\cosh (y(x))-1)=0\)

ODE
\[ (\sinh (x)+1) y'(x) \sinh (y(x))+\cosh (x) (\cosh (y(x))-1)=0 \] ODE Classification

[_separable]

Book solution method
Exact equation

Mathematica
cpu = 0.0846531 (sec), leaf count = 27

\[\left \{\{y(x)\to 0\},\left \{y(x)\to 2 \sinh ^{-1}\left (\frac {c_1}{4 \sqrt {\sinh (x)+1}}\right )\right \}\right \}\]

Maple
cpu = 0.045 (sec), leaf count = 18

\[ \left \{ \ln \left (1+\sinh \relax (x ) \right ) +\ln \left (-\cosh \left (y \relax (x ) \right ) +1 \right ) +{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[Cosh[x]*(-1 + Cosh[y[x]]) + (1 + Sinh[x])*Sinh[y[x]]*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> 0}, {y[x] -> 2*ArcSinh[C[1]/(4*Sqrt[1 + Sinh[x]])]}}

Maple raw input

dsolve(diff(y(x),x)*(1+sinh(x))*sinh(y(x))+cosh(x)*(cosh(y(x))-1) = 0, y(x),'implicit')

Maple raw output

ln(1+sinh(x))+ln(-cosh(y(x))+1)+_C1 = 0