29.26.11 problem 747
Internal
problem
ID
[5332]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
26
Problem
number
:
747
Date
solved
:
Sunday, March 30, 2025 at 08:00:18 AM
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: 5.333 (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: 35.735 (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)
\[
\left [ y{\left (x \right )} = \log {\left (\frac {C_{1}}{\sinh {\left (x \right )} + 1} - \sqrt {C_{1} \left (\frac {C_{1}}{\sinh ^{2}{\left (x \right )} + 2 \sinh {\left (x \right )} + 1} + \frac {2}{\sinh {\left (x \right )} + 1}\right )} + 1 \right )}, \ y{\left (x \right )} = \log {\left (\frac {C_{1}}{\sinh {\left (x \right )} + 1} + \sqrt {C_{1} \left (\frac {C_{1}}{\sinh ^{2}{\left (x \right )} + 2 \sinh {\left (x \right )} + 1} + \frac {2}{\sinh {\left (x \right )} + 1}\right )} + 1 \right )}\right ]
\]