62.13.3 problem Ex 3

Internal problem ID [12799]
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 : Wednesday, March 05, 2025 at 08:32:31 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.078 (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: 0.107 (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: 150.775 (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)
 

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