##### 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 = 0.598406 (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 = 0.155 (sec), leaf count = 35

$\left [y \left (x \right ) = \mathrm {arccosh}\left (\frac {{\mathrm e}^{2 x}+\frac {2 \,{\mathrm e}^{x}}{\textit {\_C1}}+2 \,{\mathrm e}^{x}-1}{{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}-1}\right )\right ]$ 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))

Maple raw output

[y(x) = arccosh((exp(2*x)+2*exp(x)/_C1+2*exp(x)-1)/(exp(2*x)+2*exp(x)-1))]