56.13.3 problem Ex 3

Internal problem ID [14159]
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 3
Date solved : Thursday, October 02, 2025 at 09:17:51 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 29

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

\begin{align*} y &= -1 \\ y &= 1 \\ y &= -\sin \left (-x +c_1 \right ) \\ y &= \sin \left (-x +c_1 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.063 (sec). Leaf size: 41

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

\begin{align*} y(x)&\to -\sin (x-c_1)\\ y(x)&\to \sin (x+c_1)\\ y(x)&\to -1\\ y(x)&\to 1\\ y(x)&\to \text {Interval}[\{-1,1\}] \end{align*}

✓ Sympy. Time used: 100.068 (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)
 

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