56.13.5 problem Ex 5

Internal problem ID [14161]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 24. Equations solvable for \(p\). Page 49
Problem number : Ex 5
Date solved : Thursday, October 02, 2025 at 09:17:53 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 17

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 21

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

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

✓ Sympy. Time used: 0.614 (sec). Leaf size: 31

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

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