23.2.58 problem 60

Internal problem ID [5413]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 60
Date solved : Tuesday, September 30, 2025 at 12:40:00 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}-2 y^{\prime } \cosh \left (x \right )+1&=0 \end{align*}

✓ Maple. Time used: 0.058 (sec). Leaf size: 19

ode:=diff(y(x),x)^2-2*diff(y(x),x)*cosh(x)+1 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -{\mathrm e}^{-x}+c_1 \\ y &= {\mathrm e}^{x}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 45

ode=(D[y[x],x])^2-2*D[y[x],x]*Cosh[x]+1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x(\cosh (K[1])+\sinh (K[1]))dK[1]+c_1\\ y(x)&\to \int _1^x(\cosh (K[2])-\sinh (K[2]))dK[2]+c_1 \end{align*}

✓ Sympy. Time used: 0.303 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*cosh(x)*Derivative(y(x), x) + Derivative(y(x), x)**2 + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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