23.2.8 problem 8

Internal problem ID [5363]
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 : 8
Date solved : Tuesday, September 30, 2025 at 12:36:13 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}&=1+y^{2} \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 29

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

\begin{align*} y &= -i \\ y &= i \\ y &= -\sinh \left (-x +c_1 \right ) \\ y &= \sinh \left (-x +c_1 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.069 (sec). Leaf size: 37

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

\begin{align*} y(x)&\to -\sinh (x-c_1)\\ y(x)&\to \sinh (x+c_1)\\ y(x)&\to -i\\ y(x)&\to i \end{align*}

✓ Sympy. Time used: 3.933 (sec). Leaf size: 14

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

\[ \left [ y{\left (x \right )} = \sinh {\left (C_{1} - x \right )}, \ y{\left (x \right )} = \sinh {\left (C_{1} + x \right )}\right ] \]