66.1.14 problem Problem 14

Internal problem ID [13790]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 1, First-Order Differential Equations. Problems page 88
Problem number : Problem 14
Date solved : Monday, March 31, 2025 at 08:12:09 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 43

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

\begin{align*} y &= \frac {x \sqrt {-x^{2}+1}}{2}+\frac {\arcsin \left (x \right )}{2}+c_1 \\ y &= -\frac {x \sqrt {-x^{2}+1}}{2}-\frac {\arcsin \left (x \right )}{2}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 57

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

\begin{align*} y(x)\to -\frac {\arcsin (x)}{2}-\frac {1}{2} \sqrt {1-x^2} x+c_1 \\ y(x)\to \frac {1}{2} \left (\arcsin (x)+\sqrt {1-x^2} x\right )+c_1 \\ \end{align*}

✓ Sympy. Time used: 0.257 (sec). Leaf size: 41

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

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