23.1.748 problem 751

Internal problem ID [5355]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 751
Date solved : Tuesday, September 30, 2025 at 12:35:58 PM
CAS classification : [_separable]

\begin{align*} y^{\prime } \left (1+\sinh \left (x \right )\right ) \sinh \left (y\right )+\cosh \left (x \right ) \left (\cosh \left (y\right )-1\right )&=0 \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 123

ode:=diff(y(x),x)*(1+sinh(x))*sinh(y(x))+cosh(x)*(cosh(y(x))-1) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 2 \,\operatorname {arctanh}\left (\frac {c_1 \sqrt {2}\, \sqrt {\frac {-{\mathrm e}^{x} c_1 \,{\mathrm e}^{2 x}+\left (-2 c_1 +2\right ) {\mathrm e}^{2 x}+{\mathrm e}^{x} c_1}{c_1^{2}}}}{{\mathrm e}^{2 x} c_1 +\left (2 c_1 -2\right ) {\mathrm e}^{x}-c_1}\right ) \\ y &= -2 \,\operatorname {arctanh}\left (\frac {c_1 \sqrt {2}\, \sqrt {\frac {-{\mathrm e}^{x} c_1 \,{\mathrm e}^{2 x}+\left (-2 c_1 +2\right ) {\mathrm e}^{2 x}+{\mathrm e}^{x} c_1}{c_1^{2}}}}{{\mathrm e}^{2 x} c_1 +\left (2 c_1 -2\right ) {\mathrm e}^{x}-c_1}\right ) \\ \end{align*}

✓ Mathematica. Time used: 4.803 (sec). Leaf size: 32

ode=D[y[x],x]*(1+Sinh[x])*Sinh[y[x]]+Cosh[x]*(Cosh[y[x]]-1)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 0\\ y(x)&\to 2 \text {arcsinh}\left (\frac {c_1}{4 \sqrt {\sinh (x)+1}}\right )\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 21.303 (sec). Leaf size: 75

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((sinh(x) + 1)*sinh(y(x))*Derivative(y(x), x) + (cosh(y(x)) - 1)*cosh(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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