38.3.24 problem 32

Internal problem ID [8291]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 1. Introduction to differential equations. Review problems at page 34
Problem number : 32
Date solved : Tuesday, September 30, 2025 at 05:21:48 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.102 (sec). Leaf size: 51

ode:=1+diff(y(x),x)^2 = 1/y(x)^2; 
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: 0.123 (sec). Leaf size: 119

ode=D[y[x],x]^2+1==1/y[x]^2; 
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: 1.486 (sec). Leaf size: 53

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

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