83.23.9 problem 9

Internal problem ID [19211]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 9
Date solved : Thursday, March 13, 2025 at 01:55:44 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 29

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

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

✓ Mathematica. Time used: 0.089 (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: 147.818 (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)
 

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