69.11.12 problem 271

Internal problem ID [18150]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 11. Singular solutions of differential equations. Exercises page 92
Problem number : 271
Date solved : Thursday, October 02, 2025 at 03:03:27 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.101 (sec). Leaf size: 51

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

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

✓ Mathematica. Time used: 0.129 (sec). Leaf size: 119

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

\begin{align*} y(x)&\to -\sqrt {-x^2-2 c_1 x+1-c_1{}^2}\\ y(x)&\to \sqrt {-x^2-2 c_1 x+1-c_1{}^2}\\ y(x)&\to -\sqrt {-x^2+2 c_1 x+1-c_1{}^2}\\ y(x)&\to \sqrt {-x^2+2 c_1 x+1-c_1{}^2}\\ y(x)&\to -1\\ y(x)&\to 1 \end{align*}

✓ Sympy. Time used: 1.466 (sec). Leaf size: 53

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

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