15.19.15 problem 15

Internal problem ID [3299]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 37, page 171
Problem number : 15
Date solved : Sunday, March 30, 2025 at 01:33:10 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 29

ode:=diff(y(x),x)^2+y(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: 0.092 (sec). Leaf size: 41

ode=D[y[x],x]^2+y[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: 150.927 (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)
 

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