30.1.8 problem example page 46

Internal problem ID [5696]
Book : Differential and integral calculus, vol II By N. Piskunov. 1974
Section : Chapter 1
Problem number : example page 46
Date solved : Sunday, March 30, 2025 at 10:02:47 AM
CAS classification : [_quadrature]

\begin{align*} y^{2} \left (1+{y^{\prime }}^{2}\right )&=R^{2} \end{align*}

✓ Maple. Time used: 0.112 (sec). Leaf size: 54

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

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

✓ Mathematica. Time used: 0.246 (sec). Leaf size: 101

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

\begin{align*} y(x)\to -\sqrt {R^2-(x+c_1){}^2} \\ y(x)\to \sqrt {R^2-(x+c_1){}^2} \\ y(x)\to -\sqrt {R^2-(x-c_1){}^2} \\ y(x)\to \sqrt {R^2-(x-c_1){}^2} \\ y(x)\to -R \\ y(x)\to R \\ \end{align*}

✓ Sympy. Time used: 4.399 (sec). Leaf size: 85

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

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