89.29.1 problem 6

Internal problem ID [24908]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 16. Equations of order one and higher degree. Exercises at page 235
Problem number : 6
Date solved : Thursday, October 02, 2025 at 10:49:24 PM
CAS classification : [_quadrature]

\begin{align*} y^{2} {y^{\prime }}^{2}-a^{2}+y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.077 (sec). Leaf size: 54

ode:=y(x)^2*diff(y(x),x)^2-a^2+y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.128 (sec). Leaf size: 101

ode=y[x]^2*D[y[x],x]^2-a^2+y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.480 (sec). Leaf size: 34

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

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